If you freeze it, I recommend freezing it in a glass container; some plastics can impart a bitter taste to frozen liquids after they’ve thawed out.

Off hand, I assume it would keep for some time. Heavy whipping cream can keep for weeks (and even months), so I would imagine buttermilk does the same, but I can’t say for sure.

Usually when you use buttermilk, it’s as an ingredient in cooking–meaning the buttermilk is heated to a temperature high enough to kill most bacteria cultures in the buttermilk–so the chances are that even “spoiled” buttermilk won’t hurt you. Best bet, though, is still to use it quickly (if nothing else, it’s a good excuse to make buttermilk pancakes).