Punic is a dialect of Phoenician, a Semitic language, related to both Hebrew and the other Canaanite languages. The language of the Phoenician colonists who settled in Carthage gradually evolved with the influences of North African languages, forming the Punic dialect. Few texts in Punic have survived and much of what has survived is relatively late.

Some vernacular Punic from about the time of Hannibal does survive from a Roman play called Poenulus. One of the characters has a short monologue, at least part of which is in Punic, and the text can be found here. Despite two relatively recent grammars of Phoenician/Punic being published in the last forty years, the academic literature on Punic is limited, though the above section that I have linked has received a great deal of attention. Most of the bibliography concerns Punic inscriptions, which do not typically contains vowels (similar to Hebrew), making it difficult to recover contemporary pronunciation.

If you want to know more, see Krahmalkov, A Phoenician-Punic Grammar, published in 2001, and as I have said, there is a significant body of work on Punic inscriptions.

There's not much I can add to the Punic half of this discussion, but there is something to be said for the Roman side. Our source for "aut inveniam aut faciam" as a Hannibal quote is Livy, writing a general history of Rome two centuries after the Second Punic War. The one thing we count on in any Greek or Roman history is that the quotes and speeches aren't authentic. If it were presented as a Carthaginian proverb or a favorite saying of Hannibal's, that would be one thing, but as it's a one-off quip, I think the credit for the phrase probably belongs to Livy or one if his Roman sources, the overwhelming majority of which are lost. Not only are we iffy-at-best on what the phrase would have sounded like when Hannibal said it, it's not particularly likely that he said it at all.

A few comments on Phoenician/Punic phonology/phonetics are warranted for this conversation:

As /u/oxfordkentuckian has already noted, Punic is a direct descendant of Phoenician. Phoenician had what we might call a very shallow orthography. That is to say, Phoenician did not render any vowels with vowel markers like Hebrew and Aramaic eventually did. HOWEVER, once Punic began to evolve and pharyngeal consonants (like ʿayin [ע]) began to weaken (considerably so by the time of Punic), various Phoenician dialects began marking vowels with consonants already extant in the orthographic repertoire. What we wind up with in various extant examples of Punic is ʿayin representing vowels like [e].

One important thing to note, however, is that we ultimately don't know exactly how these languages were pronounced (though we can make some preliminary comments on things like orthography, phonology, etc.).

Krahmalkov's grammar is ok, as already mentioned here. Zellig Harris's grammar, of course, is the standard bearer, however, alongside the work of Brian Peckham.