Carthage remained a major North African city even after its sacking. How effective was the Latinization of the general area Carthage controlled before the Punic Wars? How likely would your average resident of Carthage in, say, 50 AD, have been to speak Latin vs. Carthaginian? Would the area have been just about as "Romanized" as Gaul or any other comparable Roman conquest before the Vandals and later the Arabs conquered the area?
EDIT: Although the title says specifically language, any answers in general about Carthaginian culture would be greatly appreciated.
The language you are referring to is Punic. It is believed to have survived until around 600 AD, but quite possibly survived past that. There are inscriptions in Punic well into the Common Era though, so it took many hundreds of years for it to go extinct at the very least. For example, St. Augustine makes many references to people speaking Punic or his own knowledge of Punic, in the 400s.
No one knows for sure, though Augustine of Hippo probably had a working knowledge of it, and this was in the fourth century A.D. It may have survived until the Arab conquest in the 600s, but probably died out over the next couple of centuries.