The joint in my big finger hurts after a fall, there is no swelling or bruising, I can move it normally, is it a sprain? What should I do?

It is a sprain. Finger traumatic sprain is common. Symptoms can be the result of an acute injury. The home remedies are: ice application, NSAIDs OTC every 6 hours and gentle movement of the finger. Resume your normal activities after 24-48 hours.