I just swallowed a cherry pit. Should I do anything about it? How long will it stay in my body?

You should be fine. There is a lot of room for things to pass through our digestive tract, including cherry pits. Once it gets through your esophagus (that takes only a few minutes) it should pas through without noticing it. Well, maybe you will notice it in your stool. From mouth to toilet depends on how fast your digestion is but usually 8-24 hours is about right.