People never really left Rome: describing it as "post-apocalyptic" is IMO a bit misleading (a
real post-apocalyptic cultural transition looks a lot more like what happened to the north American native cultures in the aftermath of European contact than what happened on the Italian peninsula during the collapse of the western empire).
That said, as things like coastlines and rivers changed, settlement patterns evolved, and the post-Imperial city of Rome changed size and political importance, parts of the city that had been inhabited in ancient times did empty out and came to be dominated by abandoned ruins, some of which were quarried for their marble, others left to be buried by sediment and detritus.
I think a more apt comparison here is something like what happened to Detroit, but more so and over a longer period. A good overview using the example of the Forum can be found here: https://en.wikipedia.org/wiki/Roman_Forum#Medieval