A smoker friend of mine used to carry around one of those little camera film canisters to put his cig butts in. Obviously being careful to extinguish them fully first beforehand so the plastic didn’t get melted.
Nope, needs to be something intended to be airtight. An altoids tin would still make you stink. Film canister would work, but I would personally worry about the chemicals if I intended to finish the cigarette. Personally, I use the little tubes cannabis pre-rolled joints come in.