If I were a Roman citizen, fluent in Latin, in the time of Justinian the First, could I directly speak with a Roman from 700 years ago in the time of the Punic Wars?

You certainly could. How well you understood that time-travelling middle republican, however, would depend on your geographic origin and education.

We tend to think of "Classical" Latin - the Latin epitomized by the speeches of Cicero and the poetry of the Augustans - as standard Latin. In reality, nobody ever spoke Classical Latin. Classical Latin was an artificial, literary dialect, founded on the speech of educated Romans in the capital, but never identical to it. All spoken Latin was "Vulgar" Latin, informed by its literary cousin to a greater or lesser extent, depending on one's education.

We know little about spoken Latin in any given place or time. A few texts (such as Petronius' Satyricon) provide tantalizing glimpses. Inscriptions (and especially the graffiti from Pompeii and Herculaneum) tell us more, particularly about pronunciation (traceable from misspellings). From these traces, and from changes in "Classical" Latin, we can track broad transformations over the centuries. It is clear, however, that there was always very considerable variety of idiom between regions and classes.

We know the Latin of c. 200 BC from some helpful inscriptions and a few authors. The earliest well-preserved Latin author is the comedian Plautus, who does seem to reference contemporary colloquial conventions (and admired alliteration). Unfortunately, Plautus' linguistic exuberance (and the fact that he wrote in poetry) makes it difficult to use him as a source for everyday speech. The earliest extant prose author is Cato the Elder, whose work on agriculture has survived. To provide a sample of his Latin:

Brassica est quae omnibus holeribus antistat. Eam esto vel coctam vel crudam. Crudam si edes, in acetum intinguito. Mirifice concoquit, alvum bonam facit, lotiumque ad omnes res salubre est. (156)

(It is the cabbage which surpasses all other vegetables. It may be eaten either cooked or raw; if you eat it raw, dip it into vinegar. It promotes digestion marvelously and is an excellent laxative, and the urine is wholesome for everything.)

This is, of course, literary Latin, but in a very straightforward style probably similar to contemporary conversation. For want of anything better, we might take this as our baseline for spoken Latin in the time of the Punic Wars.

Cicero or Caesar would have expressed Cato's sentiment somewhat differently, but nothing in Cato's language was fundamentally different from that of the "Classical" Latin masterpieces of the following century. Those masterpieces - the works of Cicero, Virgil, and friends - were still the basis of Latin education in Justinian's time. Latin literature of the sixth century, accordingly, tends not to look too different from that of earlier periods. To take an entertaining example from Pope Gregory the Great's Life of St. Benedict:

...coepit ipse tremens et palpitans magnis vocibus clamare, dicens Currite, currite, quia draco iste me devorare vult. (Dialogues 2.25)

(...he [a recreant monk], shaking and trembling, began to loudly cry out, saying: "run, run [to help me], for a dragon wishes to devour me!")

Not quite Ciceronian Latin, but there's nothing here that would have been unintelligible to Cato.

Spoken Latin had probably deviated much further from "Classical" norms by the time of Justinian. It is hard to say, however, how far. The Latin of Rome had always different from the Latin of other parts of Italy, and even more from the Latin of Spain or Gaul. With the disappearance of the western imperial superstructure, and the slower but equally epochal collapse of secular education, these differences became more pronounced. Some changes seem to have taken place everywhere (most famously, prepositions were gradually replacing noun cases). But the nature of one's Latin would always have been a product of origin and education - as would one's intelligibility to a Roman from the age of Cato.

Hopefully, someone who knows more about colloquial Latin will be able to add more detail.