How should I improve the drywall under my staircase before taping?

Prefill will take care of it. Prefill only enough to effectively get tape up there, then begin buildup. Feel free to go a little heavy, sand, go a little heavy, sand, etc.

Just remember though, hot mud only dries quickly on the surface, just enough to lightly sand and recoat. It's still wet underneath, so don't go too thick per coat, and make sure you give it a couple days dry time before priming and painting.

NPM is right, but first you need to check whether the resulting corners will be parallel. Even the best tape job looks like ick if you end up with a trapezoid.

Lay a sharp-cornered straightedge on each plane and mark where it intersects with the other plane. Do this for both intersections. Then, measure between marks parallel with the staircase. If it's more than 1/4" out I'd shim as needed to straighten things up.

I highly recommend pre-bent corner tape (with the metal strip embedded). Inside corners are very difficult to get straight, even for the pros. You many need to adjust the angle somewhat.