I know about the oldest joke but what's the oldest pun? How would we know it was a pun if we don't have all the vocabulary, anyway?
The book of Kells written from the 6th through the early 9th century is said to contain puns. However I've been unsuccessful in googling them. All I can find is a reference to the puns in this history website. http://irelandroadways.blogspot.ie/2013/02/from-book-of-kells-to-synod-of-kells.html
This is a link to a site where you can view the book itself. http://www.tcd.ie/Library/bookofkells/book-of-kells/
There are puns and wordplay present in the Hebrew biblical texts that are often lost in translation. For example in Genesis chapter 3 the word for "cunning" (to describe the serpent) and "naked" (to describe the humans) are very similar words, `arum and `arumim.