The common wisdom is that Arabic, as spread during the onset of Islam, only displaced other Semitic languages but never really dislodged non-Semitic tongues. Makes sense, but I would argue it never totally displaced the Semitic languages either. People in Iraq and Morocco do not, for example, speak Central Arabian Arabic. They largely speak their original language with a thick Arabic veneer. When Mel Gibson’s The Passion Of The Christ movie came out some years back I went to see it with a group from the seminary, and I was curious if my Levantine school chums would be able to understand the Aramaic spoken in the movie. Despite the European pronunciation from the characters, they did indeed understand most of what was spoken. Levantine Arabic is pretty much Aramaic (more properly Assyrian) with a huge influence of Arabic proper.
Using this model it is quite likely that the original Phoenician tongue on Malta has survived and developed through the years and absorbed vocabulary and influences from other languages. I imagine Paul would have either spoken Aramaic with his Punic interlocutors and eventually come to an understanding, or he would have had a strained, very formal conversation in biblical Hebrew and hopefully both parties would figure it out.