Can you be a bit more specific with this question? You are asking about a very broad area of time and space.

In the Levant, the Canaanite languages (which include Hebrew and Phoenician) had largely been displaced by other Northwest Semitic languages like Aramaic and Syriac as languages of daily life by the 1st century AD.

Punic was a variety of Phoencian spoken in North Africa. It declined as a prestige language and a written language after the Roman conquest, but would continue to be spoken by the peasantry of North Africa (along with Latin and the Berber languages) right up until the time of the Arab conquest. I definitely remember Augustine talking about the continuing use of Punic by North African peasants in his autobiography (300s AD), but can't seem to find the exact passage right this instant.