I would have thought the oldest sport would be simple running races as it requires no special equipment.

Cave paintings dated 30 000 years ago show people doing archery.

Gymnastics was done in China over 4000 years ago - that's pretty old.

Any 'sport' which had a war connection would have been competitely practiced from as far back as organised warfare took place, These would have included running, horse and chariot races, wrestling, javelin or spear throwing, archery, etc.
Hunting can also be taken as the oldest sport. The kings used to go for hunting. It cannot be confirmly said about the oldest sport. But wrestling seems to have been the oldest sport known to man.