A few things about stairwells and hanging drywall. It's better to hang a rip at the top of the wall so you don't break a seam right on the level of the first floor. Best thing to do is go to the lower floor or landing and measure up to the ceiling and then figure your rip and put it at the top, with the cut edge to the ceiling. We usually go with a 24" rip, cut straight or sometimes you don't if the place isn't square, old houses are famous for that, some tract builders are too.
Second thing I highly recommend you use plenty of glue and minimal fasteners. Use the glue in the gal. tubes and run a 1/2" wide bead of glue down each stud just before hanging the sheet. We use one screw in the recess on each stud (or about 1-1/2" off a rip) and one screw in the center of the field per stud. If you're hanging 54" board use 2 screws evenly spaced in the field.
Last thing, don't land the butt joints on a stud, instead break the joint between the studs and install a backerboard, (we use 1/2" or 5/8" osb, rip a full sheet into 4' x 4" strips.) Once the sheet is screwed off on the studs, take a backer and hold it in place on the butt with half of the piece exposed, and about 2" extending up behind the course above. Screw thru the sheet into the backer every 6" and place one in the recess of the sheet above.
Before hanging the sheet, score a line down the face of the sheet 1/4" from the end, then use a sharp utility knife and cut the face back at a 45 degree angle towards the end of the sheet. We do all of our butt joints this way and you can get a flatter smoother seam which requires less mud and feathering to level out.
There's also a tool on the market that creates a recessed end on the butt joint. You mist down the butt edge and and run this roller down it to recess the end. Runs a like 3 bills I believe and I heard lots of guys swear by it on pro drywall boards like wconline. I've just got used to cutting the ends at 45's and using the backers and it works good for me.
My last boss taught me this, they do drywall in the hundreds of thousands of s/f a yr & says following this procedure in stairwells & for butt joints, according to him this cuts callbacks by 30%. I've done all my jobs this way since going into business for myself about 5 yrs now and have yet to have a call back for a crack in a stairwell, trust me it works.
As to cutting down for the stairs, I've run into this before. It's alot easier if the treads don't run all the way to the wall framing. Your better framers will leave room for the board between the stair runners. But I've remodeled alot of old houses where it's either tear out the treads or else individually cut out for each one. Real PIA & time consuming, but it's just a matter of careful measurements and taking your time. I suggest you draw it out on your board before you cut & be prepared to cut at least on sheet wrong the first time you try this.
Anyway take your drywall square & put it up against the bottom of the upper sheet already hung. Slide it back until it the bottom of the square just touches the stair framing. This is where your butt will fall. Make a pencil mark on the drywall and the stairway where this 4' point is at.
Now go back to the head of the stairs and measure the distance from the bottom of the upper course of drywall down tothe floor, deduct 1/2" from this measurement. Drywall should always clear the subfloor by 1/2" minimum
Now measure from the last butt joint in the to the edge of the riser, where it drops down to the first tread. Go back to your last butt joint and measure that same distance that you just measured from the last butt along the upper course and place a mark there.
Then you measure down from that point on the upper course to the tread, again deduct the 1/2" from the measurement. Next measure the width of the tread and add 1/4". Measure up from the second tread to the bottom of the upper course, deduct your 1/2" as before. Continue on this way until the stairs fall below the 4' level and run your full sheet on out to the corner.
Then you start all over following the same procedure with the next course down the stairs. Lots of measuring and cutting involved if you have to hang it this way in residential. I think MinConst's diagram explains the same thing but with the sheets stood up instead of hung horizontal. Like I said it's a whole lot easier to hang a stairwell if there's room to get the board between the treads and the wall studs. Then you can just find your 4' mark and the short end, snap a line between the two points and cut the sheet on the angle. Sometimes though you just got to work with what you've got. Anyway hope that helps a bit.