Define ancient. Most apocalyptic stories of the past revolve around religion. Such as nordic mythology and ragnarok or the 3 abrahamic religions which all have an apocalyptic scenario. Do you mean for Ancient Greece and Rome? In Greek mythology Hesiod suggest that there were five ages and the age the Greeks were living in at the time would be destroyed by Zeus. So yes there is an apocalyptic scenario in Greek mythology but was probably not widely discussed.

Check this out: http://www.billstifler.org/HUM2130/files/2D-003-00-apocalypse.htm