This sort of comedy was performed by Mimes in Ancient Rome, some of them became celebrities in their own right. Although their work was mostly gesture they did speak funny lines in character. Mimes were often engaged to play the deceased at public funerals. At the funeral of the Emperor Vespasian, who was famous as a tightwad, the mime playing Vespasian asked how much the funeral was costing, and then said they should just have thrown his body in the river.
(Principal source: an article in Issac D'Israeli's Curiousities of Literature)

I guess it depends on how lax your definition of standup is - we know for a fact that were recitals of comedic poetry in ancient Greece, does that count?