What's the oldest documented sport in the world's history?
Defining "documented sport" as a written list of the results of a competition, such as player rankings or time or points or a written set of rules for a competition.
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.
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 jǐao dǐ (角抵, horn butting) was used in 2697 BCE.
Wrestling is a really old sport.
Ultimately sport is a trivialization of real-life skills, most likely related to hunting -- running fast, wrestling, archery, etc. So what is the oldest sport? Probably just basic survival. The grand champion of prehistoric men was whoever could most successfully feed himself and his family.
In terms of organized sport, one answer might be Pankration. Wikipedia states pankration was practiced as early as the second millennium BCE.
The existence of age groups and judges and its inclusion in the ancient Olympic Games imply to some degree that pankration had some organization to it similar to what we would expect today from mixed martial arts (pankration's closest modern descendant).
There may be older organized sports, however.