Install drywall on narrow corner adjacent to closet door frame
When you add the wallboard to the larger portion of the wall to the opening of the door, stopping all drywall at the door opening, the short portion you have the measuring tape on could take a jamb extension. Which is a piece of wood or MDF cut to fit the space out to the new face of the drywall and that will be trimmed with the same door casing as the outside.

If you are not concerned about going into that much detail, then after the wallboard is set on the larger wall, cover the corner with a nail on corner bead and smooth over with mud, sand and paint. No need to trifle with that small of a piece of drywall. the corner bead will cover it all. If you wish after that, you could add a small piece of molding like quarter round at the inside corner of the corner bead and door jamb to help finish it out a little better.