Empty the closet of everything you can easily remove. That includes clothes, boxes, hanging rods, shelves and removable hooks. The less that is in there, and the fewer things you have to paint around or cut in with a brush, the quicker you will finish.
Tape over any metal hanging hooks or other hardware you do not want to paint. Put a drop cloth on the floor and tape it down.
Use one color for the entire closet. Having separate colors for ceiling, walls or fixtures will slow the job considerably.
Roll first. It is counterintuitive, but the roller covers better and faster than a brush. The more area you can cover with the roller, the better. Roll out the ceiling and the walls.
Place the shelves, hanging rods and any other items you removed from the closet that need to be painted on a dropcloth outside the closet and paint them. Use the roller on flat surfaces, like shelving planks, for speed. Use the brush on smaller or odd-shaped items like hanging rods.
Cut in with the brush, painting all the joints between the walls and ceiling, as well as any odd areas or abutments that the roller could not reach.
Assess whether you need a second coat when the closet is dry.