After the first piece is up, just continue around the shaft clockwise (the tallest end next) taking care to butt up against the previous piece. You'll be able to do the next two pieces easily by allowing that 1/4" margin of error. Measure the bottom, top and sides. Transfer your marks to the drywall and cut it to fit. Put it in place and if it fits, press it hard up against the already installed piece, leaving your gap to be covered by the next piece. If the gap is greater than the thickness of your drywall, you really should cut a new piece. That's not to say I've never seen somebody cheat a little. You call it if you think drywall tape and mud will do.

For the last piece you're going to have to be a little more critical on your measurements and cutting. No other piece will over lap it, so it should be snug on both sides. If you did this in the order I recommend, your last piece will be the smallest one. That way you have less material to possibly mess you up. Use your drywall rasp to finish the edges smooth. Once you have a nice fit (1/8" gap on either side is fine) fasten it (I recommend screwing).

Finishing The Drywall

I'll go ahead and describe the finishing process for those that don't already know. Get some drywall tape (or the fiber mesh if you prefer) and your drywall knifes out. Take a look at the whole project and decide if you'll want to trim out the top or the bottom or both. If you want to trim out the bottom, for example, you don't need to install corner bead on the bottom.
There is no way I can think of to 'teach' finishing drywall without showing you in person. And there is no way to 'learn' finishing drywall without doing it yourself. It takes skill, so don't be discouraged if it isn't what you thought it would be. Take you time and fortunately it's a small project so you should be able to botch it up too bad. ;)
Get some all purpose compound and apply it to a corner. You want maybe an 1/8" thick layer about 3 inches from the corner both ways. Your paper will have a crease in the middle. Cut a piece to fit the corner and fold it on that crease. Start at the top and press it down all the way to the bottom. You want to get the corner in straight and without bubbles and folds. Make sure you use your knife in one direction only. For this particular install, you won't really be able to wipe up, just down.