We have the slang for "buttocks" on one hand, and the archaic term for a "donkey" on the other.
Is this pure coincidence? Is there some connection that links the two terms -- and if so, which "ass" is the original?
As with all etymology questions, the best place to look is in an Etymology Dictionary. Etymonline is a handy online one.
A search of that site for the etymology of ass turns up both definitions. The one meaning "buttocks" is an americanized pronunciation (and then spelling) of the word arse. They say that such a pronunciation goes back at least to the 18th century.
The definition meaning donkey is much older, going back at least to Middle English and ultimately deriving from latin asinus.