As long as there have been people, there has been wrestling. Nearly every culture has some form of it used for contests internal and external to the group. Wrestling is one of the most basic sports, and is probably the oldest.
In Egypt, the fifteenth tomb of Beni Hassan has a large wall depicting wrestling techniques. At this point we're talking about roughly the 19th to 20th century BCE.
We have similar evidence of wrestling in older tombs such as Petah Hotep (aka Ptahhotep). It is specifically an ancestor of Kirkpinar, Turkish oil wrestling:
The oldest known proof of the existence of oil-wrestling in Ancient Egypt is found in limestone from the tomb of Ptahhoteb near Saqqara from the fifth dynasty (about 2650 BC) from the same period as the Chafadji-bronze.
In China specifically, there is some evidence that something related to (but probably very different from) modern shuai jiao wrestling called...