How can I insulate a bedroom closet?

I would try either sheet Styrofoam, and or the quilted aluminum foil that comes in big rolls. Lowes and Menards sells that, and I have used moth methods for various insulating jobs at my old farmhouse.

If you use sheet Styrofoam or any similar insulation it must be covered. It is against fire code to leave it exposed.

put your insulation on the cold wall and make a false wall that will make the closet warmer

Generally closets are not heated or connected to the home heating system, so even with insulation they still remain cold. I assume in the spring you will have blown in insulation installed in the walls from the outside. This will help in reducing the cold, but not completely. My only suggestion is to leave the closet door slightly open to let the room air warm it up. If the door/doors are solid you can get some heating return air grills, cut openings in the bottom of the door/doors, and mount the grills to let air circulate.

Line with Polystyrene sheets.