As long as the door closes tightly, weatherstripping should be able to prevent air infiltration. Is it an actual custom size door, or is it more like a hole with a board on a hinge covering it? Without a real door, it might be hard to get the door to close tight enough to make the weatherstripping effective. You could probably get a stock solid core door for the width and cut it to length if you want to replace the door.
I've also seen zippered insulated coverings for attic hatches with pull-down stairs, but they might also be able to work for a door if it swings into the closet rather than into the attic.