If you want it you can make it happen
On the inside of the cabinet add two blocks of wood that you glue to the back of the cabinet. Position one just below the top of the cabinet, the other one position just below a shelf.
To mount the cabinet you will be putting screws that embed into those blocks. The screws heads will on the inside wall of the closet. In order to locate everything you will need to take measurements of the distance between where you want the hole locations and make a template or just put some masking tape on the closet wall and lay out those holes and also mark the edges of the cabinet location on the tape. Now drill the holes in the closet wall. Then with an assistant hold the cabinet in position, drill short little pilot holes into those blocks in the cabinet and then drive in the screws. It will not fall
of the wall
If you are concerned about the door hinges coming loose take out the screws and put a little super glue under the hinges. You can do the same thing for the mounts for the catch. I like the Gorilla Glue brand of gel superglue for that purpose, it is a rubber fortified formula that is shock resistant. It bonds well to metal and wood.
If the door gives you trouble with coming open put in a couple of brass screws and use little cloth covered hair tie rubber band stretched between them to hold it closed while under way.
Sure the glue on vintage pieces can get brittle and fracture. If that happens it is likely hide glue and can be removed with scraping and a bit of heat then just glue it together with clear flexible caulking that is rated as an adhesive. That stuff holds up to all kinds of movement without ever letting go.