The only cases I worry about cleaning before storing are black powder cartridge cases. Unless they landed in the mud or something, my smokeless powder cases just get tossed into storage and cleaned later on, just before I reload them. The fact is, even later on my smokeless powder cases might not get a trip to the tumbler before I reload and fire them again. I'm not all that particular about having bright, shiny, new-looking cases. Most of the time my cases just get wiped off with a relatively clean rag. Of course their primer pockets get inspected, and sometimes cleaned, before I put new primers in them.
I think there's probably a lot of handloaders who use the same storage containers as I - old coffee cans with masking tape labels on them. Those gift tins that you can buy at the Dollar Store for a buck apiece around Christmas work well too.
One thing though - don't store your cases in a damp basement in the original cardboard boxes they came in. They'll get some kind of green gunk all over them. Ask me how I know.