To properly store a shotgun long term you should completely disassemble it and individually number each part, then metal parts oiled and put in a vacuum-sealed bag and wood parts waxed then put in a climate-controlled locker.
Although I'm "sort of" kidding, military storage of surplus arms often involves a bath of cosmoline, then a plastic (or wax paper) wrap before shelving.
You should probably define "long-term". Generally, when I'm done shooting I'll give a gun a good clean and lube then into the safe or case, muzzle up because it is convenient. The storage area is somewhat climate-controlled with aids in preventing rust. If the gun didn't see a lot of rounds, say only 50-100, and it doesn't look dirty, I'll just give the barrel a pass with an oily patch and maybe a drop or two here or there and a wipe with oil. If I were to store long term say for collecting purposes, I would clean and oil, then wrap the gun in plastic shrink-wrap with paper towels down hill from the receiver/bolt/trigger to soak up any excess oil, then I would put it back in the original box and shrink-wrap it and probably store it horizontally like the mfg does. But then you go into a gun store and their guns are vertical on the rack so who knows...
Note that oil will "settle" and could possibly discolor a wood stock of a gun stored muzzle up. So can some solvents/cleaners. You could also see what the mfg suggests and you will probably find out that everybody has their own "special" way of storing a gun.