In what order should I play this on piano? Where exactly metronome tick should appear? 4&1 - 3 - 2 begin with metronome tick 4 - 3 - 2&1 end with metronome tick 4 - 3 - 2 - 1 start or stop with metronome tick
The initial 8th notes in the left hand are like a written-out, slow, arpeggio.
The arpeggios marked with wavy lines need to be faster than those 8th-notes, otherwise they will distort the rhythm so much that it "falls apart" and listeners can't follow the beat at all, but the 8th-notes are a strong clue about how to play the arpeggios IMO.
If you interpret the notation literally, it would mean the second note of the two-note arpeggios goes on the beat, at the same time as the right hand note. However I would be inclined to play both notes of the arpeggio before the beat, following the pattern of the written-out 8th note "arpeggio". Otherwise, the two-note arpeggios sound a bit "lame and weedy" to my ears. For the three-note chords with arpeggios, maybe it makes more sense for the third (top) note to coincide with the right-hand note, on the beat, following the overall pattern.
For scores like this, you have to make up your own mind how "professional" the notation really is - does it literally mean what it says, or did the person who made it not have much in-depth knowledge, but just took whatever their notation software did by default as "good enough"? Sometimes, identifying which notation package they used (from give-aways like the music and text fonts being used, and more subtle clues from the default layout of the complete page or the note spacing) can be a forensic clue as to what the writer really meant. Even if the writer "knew" the difference in meaning between the arpeggio lines as written, and lines that spanned both staves, some notation software is more "friendly" about creating lines across both staves than others! It doesn't make sense to take what is on the page completely at its face value, unless you feel sure that the person who created it (whether the composer, or someone else) fully understood exactly what it meant.
For me (as a "nit-picker" about music notation!) the fact that the quarter-note rest in the pickup bar in the right hand is in the wrong position rhythmically (it should be vertically aligned with the first of the two 8th-notes in the left hand) is a clue that "this music isn't engraved to a professional standard, and/or it's not engraved with professional-standard software, so don't take everything else you see on the page literally" - but you might choose to take a different position from me over how much you read into "details" like that.
In this case, I think the rhythmic integrity of the RH melody should prevail, the arpeggio should lead into it.
If we can not be completely ruled by the 'tick' and play a little more freely, it wouldn't be unreasonable to start the arpeggio on the beat, giving emphasis to the melody note by delaying it a little.
Not 5 times in 8 bars though! Arpeggio before the beat, melody on it. Final answer.