[The Might]

Makes a lot of sense since Hobbits were known as good archers and could aim pretty well as shown by Bilbo when throwing with stones at the spiders.
I think that the type of sport would depend on the region.
Hobbits would probably have sports similar to our own considering what is said in the books.
Rohan would definitely have horseriding.
Elves might have archery contests, just as other areas.
Swordfighting wouzld probably be quite widespread throughout Middle-earth.
Orcs would perhaps like gladiatorial type of games where the winner takes all.

Anyway, except quiet and peaceful areas like Hobbiton, sport in other areas would probably often have much in common with war. Sport might be used a way to stay fit and practice, so that you would be ready in case of a real battle. It could also create a competitive spirit between the people in the area and each would try to become the best, thus having a positive outcome.