First of all, I think it's important to keep in mind that harmony is written for something. I'm going to follow the sort of academic rules of part-writing in my answer, but if this were for "real life" purposes, it would be worth subjecting a lot of those assumptions to the test of what makes the best music (and the most idiomatic music for whatever you're writing for). I think there could be a lot of musical merit, for example, to using only two voices in the pick-up bar, and possibly even to not always treating the two voices given as "bass" and "soprano."
Within the bounds of formal part-writing, though, the place to begin has to be harmonic analysis. That answers the question of quarter-notes vs eighth-notes right away: all of the eighth notes in this excerpt can be analyzed as passing tones, so they aren't a part of the functional harmony. The harmonic rhythm here is definitely the quarter-note. Of course, once you've written a harmonization in quarter-notes, you may want to add non-harmonic tones to your lines for musical interest, or you may want to use them strategically to smooth out voice-leading — this comes back to what the arrangement is for. (If we were going outside of academic part-writing, you might even want to consider using a primary harmonic rhythm longer than the quarter-note, but the quarter-note will do well for study purposes.)
When that's established, the first thing you'll want to do is harmonize the E-minor
i chord on the first beat of the first full bar. That
i chord is, from the perspective of functional harmony, the real "landing point" for the beginning of the piece. The function of the pickup bar is to propel us to the
i chord on the downbeat, so the best strategy is to first figure out where we're going, and then work out the most elegant way to get there.
Of course, there is no one right answer for how to harmonize that
i chord, but I'm going to go with
e2 e4 b4 g5 (if you enter that into Wolfram Alpha you'll see the notation).
Once we've established that
i chord on the down-beat, we can go back to the pickup bar and and analyze how it is propelling the music forward to the down-beat. Since we know that we're headed to a
i chord in E-minor, the best analysis for the harmony of the pickup bar seems to be as a
i^6 chord (i.e. a
i chord in first inversion). If you exclude the second eighth-notes of each pair as passing tones, you'll also see that the soprano and bass voices are in fact exchanging pitch-classes from the pickup to the down-beat.
The goal, then, should be to fill in the harmony in the
i^6 chord in a way that:
- is unintrusive; and
- helps to keep forward momentum toward the down-beat (
How to achieve those goals is again a matter of taste, but in my example solution, I'd again give the tenor
e4 and the alto
b4. That way, the alto and tenor will stay on the same note from the pick-up to the down-beat, which lets the movement in the bass and soprano shine through while still establishing the four-voice texture.
Hopefully this example both helps you through the pick-up bar and demonstrates the strategy to use in the rest of the piece: start by identifying (and harmonizing) the most structurally/functionally important moments, then fill in the details to achieve smooth, elegant, effective voice-leading.