Note: this answer uses the ABC notation to show some examples, but there may be some browser issues (like this); try to force refresh the page (if you're on Chrome, use CtrlF5 or ⌘⇧R on macOS) to see if it properly loads them, alternatively click on the "Image link" provided below each example.
That part is really ambiguous. It's not impossible to read or play, but it leaves a lot to the experience of the musician.
Note lengths, rests, ties (or slurs?), and even the instrument indication of "Cymbal" in singular form: a lot of things are left unsaid.
When unsure, asking more experienced people may help, starting with the conductor. If they can't (or won't, possibly for didactic purposes) give you specific instructions, listening to recordings always helps, even if they are not from the same arrangement. Even looking for scores from other versions and transcriptions may help, at least to understand how others have decided to write that part. Studying the full score can also be very helpful.
The problems with percussion parts (at least, some of them)
Note that, statistically, most percussion parts are terribly written.
Learn to live with it: a good part of the life of a percussionist is to understand what composers really wanted, as opposed to what they actually wrote. While more standard instruments have a very well established notation technique that is extensively taught in composition courses, that's rarely the case for percussion. Sometimes, composition teachers just tell "let the percussionist care about that", which, controversially, is also a good advice (I've seen parts written by "too precise" composers that cared too much about the form rather than how the result could be possibly portrayed).
Even with experienced and famous composers (or renowned editions), you can easily find parts with completely inconsistent notation, note/rest lengths, strange marks, etc. Sometimes, they just do not make any sense at all. I recently revised a part that had a diminuendo hairpin for a final note of the Xylophone, and it wasn't rolled.
The reason is that composers usually know very little about percussion (or assume they know enough), including specific techniques of each instrument and the expectations about their notation.
Even the naming is problematic. For instance, you say you're playing cymbals and I'm assuming you mean "pair of cymbals", which we can also find noted as:
- a 2 cymbals, hand cymbals, clashed cymbals, crash cymbals (English, even from non native speaking)
- piatti, piatti a 2, cimbali, cinelli (Italian)
- Becken (German, ambiguous as it's both singular and plural)
- cymbales (French)
Sometimes, we also find it in the singular form ("cymbal", "piatto", "cymbale"), which is really ambiguous. Not to mention languages uncommon for music notation, like Spanish.
Interestingly enough, that's exactly your case: your part says "Cymbal".
For band and marches, it's usually implicit that they meant a "pair of cymbals", as it's based on the tradition (coming from the Turquerie and its evolution in the Western music); but that may not always be true, especially for marches written/arranged for full symphonic orchestras: after acquiring aspects of foreign ethnic music, composers and orchestrators also became more aware of their characteristics and potentialities.
Deciding whether to use one or the other depends on experience and, possibly, a musician's choice (including that of the conductor).
From what we know, the arranger may have chosen to actually use a single cymbal, possibly to be played with mallets. I doubt that, but the very recording pointed to above in the comments also seems to be using a suspended cymbal for the section between bars 22 and 34 (most certainly using a wooden stick).
Understanding, interpreting and managing durations
First of all, remember that percussion instruments may have different characters: while they mostly are used for rhythmic purposes, they also are often used as a "sound effect", for which they timing may become more "fluid", not only for their effective duration, but also for their "attack" time (consider a crescendo roll on a suspended cymbal).
Now, the problem with the duration of notes in percussion instruments is that they are almost always virtual, because most of these instruments don't have a specific duration or cannot have it with single hits or strokes. Most of the times, the notation has to be considered with three aspects in mind:
- the musical intention;
- the simplicity of reading/writing;
- the reference pulse of the meter;
It's also worth noticing that, until a few decades ago, percussionists were generally not particularly skilled musicians, and some parts were (and still are) often written with less care than others because who wrote them knew that writing more accurate parts wouldn't have mattered.
Another important aspect is that, for percussion, there's almost no difference between slurs and ties. Due to the physical nature of most of those instruments, there can be almost no difference; we mostly have single repeated strokes at best: we cannot physically "slur" them. An accurate and aware notation may be performed in slightly different ways by an experienced performer, but those differences are almost unnoticeable, and have to be enforced by other aspects (such as careful dynamic lines, extremely precise time/tempo management, and proper awareness of musical phrases).
For instruments with almost no sustain (like a snare drum), the length of a single note is irrelevant: give the following to a snare drummer, and you'll always hear the same result from each bar.
K: perc stafflines=1
B2 B2|B z B z|B1/2 z1/2 z B1/2 z1/2 z|B1/4 z3/4 z B1/4 z3/4 z|]
Obviously, the first two are the most preferred choices, but the second one is probably the best: it makes sense (there's no point in a full half note), it's easy to read, and it also takes less horizontal space within the staff than the last two, allowing better printing layout (less pages) and reading (less page turns) in the long run - also for the score.
For instruments with longer sustains, it gets tricky, especially when dealing with rests and syncopated rhythms.
Imagine that you want to write a cymbal part for a march, where they play quavers on the upbeat along with the snare, following the bass drum on the downbeat. You, as a composer, will most certainly write this, and the player probably never damp the cymbals within the same musical phrase:
K: perc stafflines=1
z B z B|z B z B|]
In reality, the part should technically be written like one of the following two bars:
K: perc stafflines=1
B B - B B ||-B B2 B-|]
That's clearly awful: both cases are unreadable, especially considering that a part like this may go on for dozens of bars (meaning that it's also impractical for the writer, other than a nightmare for the reader).
Now, consider the case in which, in the same piece, you read this as the cymbal(s) player:
K: perc stafflines=1
B z B z|B z B z|]
How would you interpret that? Should you damp the cymbals or not?
Another similar problem happens when you have the typical grandiose cymbal hit that is intended to last even more than a single bar. As said above, we also have to consider whether a percussion sound is an effect, or part of the rhythm. But there's no point in writing the "real" duration of that cymbal clash: is it a bar? Two? Three bars and a quaver? As a cymbal player you have little control over the duration of their sound: if it ends, it ends. At most, you can eventually stop it, no more than that.
In case the effect has to last "for some time", it's normally written as a note (no matter the duration) with a slur that isn't connected to any other note:
Alternatively, the technique text "l.v." ("lasciar vibrare", let it ring) may be added. This is opposed to "Ch." (or "Choked") text, which can also be written using a small apostrophe-like symbol, or, sometimes, the "staccato dot" on the note.
Still, the duration is up to the taste and intelligence of the musician and/or the request of the conductor.
Finally, the case at hand
Assuming that that part is actually intended for pair of cymbals, and after listening to the example linked above, here are my suggestions:
- the first hit may last longer than indicated, as it's more of an "effect" than a rhythm; making it last as written may be valid too: in that case, I'd probably damp it "softly", not abruptly at the start of the second beat of the bar;
- notice the difference in duration with that first note and the others from bar 5; assuming that the part was written with awareness (and considering the mp dynamics) it probably means that you could softly damp those crotchets so that the sound cannot be heard on the second beat;
- I wouldn't care too much about the rest at the end of bars 8, 12 and 16: almost nobody would notice any difference even when you're playing in solo, and, unless you're very skilled with cymbals, trying to do that may even be dangerous (and pointless) as such dynamics: almost nobody would hear (or care) about the difference, but they will if you delay the following hit because you were too worried about the dampening;
- for the same reason, don't care too much about the quaver rests before 22;
- passages like in bars 8/9 and following are fundamentally similar to the other you circled at 40/41 (and, actually, 36/37);
Now, the problem with the last point may consider another aspect related to the cymbal skills mentioned above.
There is another cymbal technique involved. I don't know the English term, in Italy we colloquially call it "far friggere i piatti" (make the cymbal fry); it differentiates from the typical "hit and run" cymbal clash in the way that the borders of the cymbals are still close enough to let their vibrations hit each others (but far enough to not stop dampen themselves, as in a choke strike), which gives a characteristic sizzling (frying) sound which is somehow similar but also different than the normal cymbal sustain vibration.
I would probably try to use that effect in those bars, especially because they're followed by an accented note which will emphasize what preceded it, almost like some sort of crescendo.
There's no absolute "right" way to do it
Still, it's a matter of taste and style.
Remember that it's music; it follows rigid and strict rules, including the fact that there cases in which rules cannot be that rigid and strict. You may even choose or need to do something completely off (or "wrong"), but that may make sense for the purpose of the performance. For instance, if you've seen some video from Igudesman & Joo (who are perfectly fine music professionals), you'll probably understand that.
If you're still in the beginning phase of your learning, the suggestion is to try and learn from your attempts, including the wrong ones. Do experiments on your own, analyse them and think about what you like or dislike more, then try to mindfully apply all that during rehearsals in order to hear and understand if it also makes sense in the overall result (rehearsals are for that too); eventually ask the conductor about the alternatives (during the rehearsal, or following them, privately): if she/he is also a teacher, it's possible that their "just listen to recordings" was their way to tell you to "learn how to learn", and your personal research on the matter alone will be enough for them to finally tell you what they'll actually want, from the ensemble and from you, possibly including "do it as your taste tells you".