There is not a lot of good evidence for Carthage in the period from the end of the 3rd Punic war to the founding of the Augustan veteran colony on the site at the end of the 1st century BCE. Despite it often being repeated, the lands of Carthage were not sown with salt. The city was destroyed by fire and sword at the end of the third Punic War, and Scipio thought of the legendary destruction of Troy as he watched her burn (Polb. 38.24).

The administrative responsibilities of the new province were moved to Utica, up the coast, where there was (for a time) a navigable river outlet. Utica had been a smaller satellite city of Carthage during Punic times. No doubt the move of the "capital" was meant to punish the ancient city symbolically, but there were also economic reasons (the river and harbor). It is possible that the Romans deliberately destroyed the great harbors at Carthage during this period, but there is no archeological evidence of it. This period of the city is not very well-known as far as archaeology, but there are scattered indications of occupation. In other words, the place was not completely deserted. Your Punic everyman living in Carthage after the wars would dwell in a ruin. Many of the houses and such certainly survived, but it is probably safe to assume that most of the major civic structures of the Punic civilization were razed (temples, public spaces). Your guy would need a reason to continue living there, and a source of livelihood. Perhaps he is a scavenger, or an expert in locating buried valuables in the rubble, or simply subsists on local fishing and hunting. The civic structure, however, must have been pretty thoroughly unraveled, and there was no impetus among the new masters (the Romans) to rebuild.

In 122, a Roman politician called Gaius Gracchus proposed the founding of a new colony atop the ruins of Carthage, but that project was aborted for political reasons (it would have been used to provide land to the poor). 80 years later, Julius Caesar revived the idea of a colony there, this time for discharged veteran soldiers. He was killed before the project could be fully invested, but his predecessor Octavian, who became the first Roman Emperor Augustus, completed the idea and founded Colonia Concordia Iulia Karthago. This was just one of dozens of veteran colonies founded by Augustus across the Mediterranean, settling tens of thousands of soldiers into new lives.

If your Punic survivor persisted in living at Carthage, and his sons, and grandsons, and great grandsons, they would have witnessed a profound change upon the arrival of Augustus and his colonists. The Byrsa hill, which had been the center of Carthaginian religion, was completely razed at the end of the 1st century, and the top was leveled. The dirt and debris from the top was pushed onto the sides of the hill, and the Romans constructed massive retaining walls to maintain the shape of the new "plateau." On top of the flat space on top, they built a new Roman forum with Roman temples, courts, and public spaces. The alignment of the city streets was redefined, and the entire countryside of Tunisia was set to a Roman grid and parceled out to colonists. If there was a substantial population of old Punic survivors dwelling at the site, they almost certainly lose all they had or were moved into new, designated "areas" (ghettos).

This does not mean that Punic culture was completely erased. In fact, the archaeology of Carthage and Tunisia shows a persistence of "Punic" influence even into the 2nd century CE, two hundred years later. The Punic god Baal, for instance, became associated with the Roman god Saturn, and we find evidence of cult observance of "Saturn" in North Africa all the way into Late Antiquity and the coming of Christianity.

Some good basic sources for Carthage:

Hoyos, B. D. 2010. The Carthaginians. Milton Park, Abingdon, Oxon [England]: Routledge.

Lancel, Serge. 1995. Carthage: a history. Oxford, UK: Blackwell.