From what I understand, stand-up seems to be a fairly modern thing. (humorous monologue that is not considered part of a play; usually personal stories meant to be funny)
Not entirely too modern. In China, the tradition of a person standing there and telling a humorous story, or bantering with a second actor playing the 'straight man' is quite old, dating back to at least to the late Qing dynasty.
Here's the Wikipedia link, if you wish. Similar forms of performance in the rest of Asia also exist, I believe the Japanese form is called Manzai.