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.