Install drywall on narrow corner adjacent to closet door frame

I'm looking to finish the inside wall of my closet by putting up drywall. Most of the closet is finished, but the backside of the wall with the sliding door on it isn't.

There's a vertical wood stud that sticks out ½ inch past the door frame, and I'm not sure I can reasonably fit drywall onto that narrow surface.

Photo from the inside, looking up and towards the room (the phone cord is unimportant and will be removed):

I sketched the area for my own reference:

How can I finish this? I thought I might install drywall onto the larger surface, then wrap corner bead around the new drywall and directly onto the wood stud. Is this reasonable? Is there a better option?