It's a big question. I'll stick to the money aspect. I had a quick look around for "Roman Republican border customs" but could not find anything informative that was low-hanging fruit, and I honestly don't know the answer. I know border customs existed in the ancient world as a concept. There is the famous anecdote about Alexandrian ports inspecting (and seizing!) incoming ships for literary works, then returning copies of the books. I just don't know enough about this topic. I am confident that ships did not arrive at Ostia just willy nilly with no sorts of customs, but I'm unable to find reliable information on it right now. But anyway, on to the coinage.

It's interesting that you picked Egypt, because in the 1st century BCE, this place was unique among the Mediterranean states. Though some other Greek states continued to mint their own coinage, for the most part by the middle of the 1st century BCE the control of precious metal bullion and the minting of coinage had fallen almost exclusively to Rome. We don't know why for sure, but it must have involved a combination of obvious factors (Rome's imperialistic activities, the expansion of the Roman banking system into the provinces, etc; see Hollander on this). We see the slow but steady disappearance of other currencies from the Mediterranean economy after the middle of the 2nd century BCE, which coincides with Rome's effective dissolution of the Macedonian state and simultaneously the Carthaginian state. This included their coinage, but also Gallic and Spanish gold coins and the coinage of a handful of prominent "independent" Greek cities like Massilia. Except for Egypt.

The Ptolemaic state had inherited the sophisticated banking practices of the Late Classical/Early Hellenistic periods, just as Rome had, and had preserved them in what used to be called a "closed" system all the way up until the dawn of Roman interference on the Nile in the 1st century and indeed well beyond. While non-Roman coin types steadily disappeared from the rest of the Mediterranean, in Egypt the old Macedonian tetradrachm persisted as the dominant (or even exclusive) currency all the way until 296 CE. Egypt's banking system by the 1st century was arguably the most robust in the entire ancient world: not only were all the basics of modern banking in place (accounts, credit, basic cheques, risk management, etc etc), but Egyptian bankers could transfer money between accounts, were creating joint maintenance accounts with other bankers (Egyptian and Roman), and were honoring basic cheques broadly (though there was not such a concept as cheques from strangers). The system was robust and was used even for small payments. One caveat: because of Egypt's relationship with papyri, we know a lot about Egyptian every-day accounting, and the availability of evidence surely skews our perception. Some (like Rathbone) argue that the only special thing about Egypt is the abundance of evidence, and that banking as probably robust in many provinces, especially former Macedonian states.

The Roman banking system was of course also quite advanced by the 1st century BCE. We should probably transpose most if not all of the Egyptian mechanisms onto the Roman system, if only we had more evidence for the nitty gritty of Roman practice. In short: your traveler would have no troubles. In Rome there were a number of banking-related professionals who performed dozens of roles. The argentarii were general bankers who ran businesses pretty much identical to a modern bank: they maintained accounts, they underwrote risk, they lended, the borrowed, the speculated, etc. Along side them were the nummularii, who (among other things) tested coinage and exchanged it (both between currencies and denominations, and also from bullion or in kind). Your traveler would have no trouble getting it exchanged upon his arrival in Rome (for a small fee, of course—bankers are bankers, then and now). There was an apparatus of exchange even in more wild places: for instance, Cicero, while governor of Cilicia, found himself in possession of a large sum of cistophori (a Hellenistic coin type still persisting in the East in the 1st century BCE), and tells his friend that by the exchange (ex permutatione) of this coinage into Roman coinage he could maintain his accounts (see *ad Att. 11.1.2).

A note about your traveler: we must assume he was a person of means. This means that he would not be traveling with a chest full of coins, especially not between two sophisticated banking areas. If a merchant, the majority of his value would not be in coin, but in goods; if a diplomat or envoy intending to bribe some Roman senators, he would almost certainly travel with bullion, not coin, or on transfer (see next). Monetization was the engine of commerce and the highway between different commercial spheres. If for some reason he needed to transfer a large amount of value, he might arrange a transfer, what Cicero confusingly also calls a permutatio, which would, as in a modern system, shift that account value from Egypt to Rome via writ and not via the physical transport of thousands of pounds of actual metal in a fragile ship. This could have been accomplished between Alexandrian and Roman argentarii, or via specifically Roman argentarii already in Egypt (which surely they were). I'm unable to find a specific example. Cleopatra traveled to Rome and remained there for some time, but the sources do not talk about how her wealth was transported with her. She, of course, had the means to physically transport anything she wanted, while our theoretical traveler would probably not want to undertake the extra expense or risk.

For money in the Late Republic, see the aptly-titled work by Hollander, Money in the Late Roman Republic (Leiden: Brill, 2007). Sitta von Reden also has a nice chapter about the advent of Roman monetization in the Med. in the Cambridge Companion to the Roman Economy (2012, ed. Walter Scheidel). It is a complicated topic, but both do a good job of breaking down what exactly coinage was in the ancient world, and what it did or did not do.