Is there any evidence that thinkers in ancient Rome or Greece postulated what life would be like thousands of years into the future? What did they say about how they expected humanity to be living in our modern times?

Yes, they postulated, but there's an important difference between them and us to keep in mind

Western culture have inherited Christian idea that time is a line. We live in the present, there's a unchangeable past behind us, there's an unknowable future. In Christian terms: as humanity we are between Creation (of time itself too) by God and the end of times, apocalypse, judgement day yadda yadda. This is our view on everything: we are born, we get old, we die. A technology is invented, is developed, became obsolete and so on. There's no come back.

But for Romans and other people in the ancient Mediterranean time is a circle, history will always reset and start again. So a Roman looking at the far future of humanity would see just other civilizations, other languages, other cultures, but in short something very similar to Rome because there's no possibility for radical changes if history start again from zero every time.

In "Somnium Scipionis" (Scipio's dream) Cicero write a fictional dialogue between Scipio Aemilianus and his deceased uncle Scipio Africanus. The uncle's spirit tell the nephew to not tend to mortal glory because it's limited by space and time. By space because even Rome cannot conquer the entire world (Cicero believe there was a huge wall of fire on the equator), by time because there will be, as there was, a kind of natural event that destroy everything of the old world, even ruins and memories, and the cycle will start again.

Hesiod in his poem "Works and days" write about five ages of Man (Gold-silver-bronze-heroic-iron), each one with different type of human beings, too.

Most of all, keep in mind that "progress" is a very recent idea. The evolution of mankind in the last two centuries is far faster than previous 5000 years. Your and your grandpa's life are much different than an ancient Egyptian and a Pilgrim life.