The religions of ancient civilizations often had an eschatological perspective incorporated into their religious views -- Ragnarok, Armageddon, the age of Shiva -- but do you want to limit your question to what might be called "secular" perspectives, e.g., what if Athens is sacked by Macedonian invaders and its inhabitants shipped off into slavery, rather than what if Shiva destroys the whole universe than then is recreated by Brahma who is reborn from a cosmic lotus flower?

Certainly you can see in late Roman texts some acknowledgment that the Roman Empire was facing big problems. No one, as far as I know, really predicted "Well, the Western Roman Empire will collapse in the 5th century, while the Eastern Roman Empire will survive about a thousand years more." Vegetius in De Rei Militari, for example, talks about the problems facing the Romans from a military perspective, and his proposed policies (e.g., renewed emphasis on old Marian-style heavy infantry) for remedying the situation. Augustine in City of God, attempts to refute the idea among pagans that the Roman Empire's troubles are effectively a result of the Romans abandoning the old gods for Christianity -- while also stressing that the secular fate of Roman civilization is of secondary importance compared to God's Kingdom.