Yes and no.
Yes because there were certainly denizens of the Roman Empire who were harping on about the end of days (think about how many early Christian writers were inherently eschatological).
No because life before and after 476 CE continued on. This in fact, was the essential conundrum that Augustine tried to bridge after the Sack of Rome in 410 CE. He had to address the fact that to many Romans, it felt like the end of the world, but he wanted to highlight it was only the end of A physical world (and thus the physical world continues), and that we should be looking forward to the end of ALL physical worlds while looking forward to the arrival of the spiritual world.
Yes and No because this question reflects the inherent problems of Predicting The End: Things end and keep going at the same time (consider the fall of the 3rd Reich and the Soviet Union). And there are always people predicting the end, and when an end happens, they would've turned out to be right, but this ignores all the other times people predicting the end turned out to be wrong.
So with this in mind, was someone who was consciously talking about, or feeling the End of Civilization, the equivalent of a perpetual doomsday forecaster (like Christian priests who've been trying to predict Jesus' second coming for the last 2000 years), or someone predicting the end of their political state, of which life will continue on afterwards?
It's very hard to disentangle the two, because we conceive of the Roman state as separate from human civilization. It's not so clear the Romans did, whether pagan or Christian.
But this raises a problem for us and your question. I am presuming by the grammar of your question you are asking about the former, did the Romans note that the end of their civilization (but not the end of the whole world) was happening? My point is simply, if we can't disentangle the two, we have a hard time answering that question, because then your question becomes "did citizens of Rome feel that the world was ending?"
To which I can confidently answer, "yes," but the separation between one physical world and all physical worlds clouds the answer, because I surmise you want to ask a question equivalent to "how did the citizens of Nazi Germany or the Soviet Union feel when their state was ending" (because we surely know that they knew their state was ending, even though they knew the world would go on).
Just some food for thought.
May I add on?
Was Rome indeed sacked on several occasions, and if so, could those reasonably have been considered significant "falls" of one nature or another?
After such events, I would think life as usual for Roman citizens at the very -least- changed in various ways, and the conscious or unconscious idea that 'Rome was eternal' would have taken a pretty memorable beating. Would you say that's correct?
Were there indeed "barbarians approaching the gates" moments that ordinary citizens had to somehow deal with, perhaps over the course of weeks, months, or even years?
The fall of the Roman Empire was a very gradual process spanning nearly a century, when the Emperor Diocletion split the empire to four teritory each having their own government effectively turning the Roman Empire into a Thetrachy to avert the third century crises, Rome had bassically ended its own political and cultural significance to the empire. The crisis was a fatal blow to Rome, the combined Barious invasions, political instability, and civil war spun the empire into a state of dissaray, and Rome followed suit. After the Tetrachy the city's population decreased gradually, in the late fourth century it estimated Rome had only a quarter of a million citizen, then West and East divide happen, the Gothic Invasions reappeared and the rest is history.
So I guess, the people living there had the right idea when the news of barbarians and civil wars were looming and get the hell out of there, But in the bigger scheme of things the common citizen obviously have no comprehension of what was going in the vast expanse of the empire, not like anybody had any.