First, and to me, most obvious, would be language change. Like all language, Latin was subject to change over the centuries. For example, Shakespeare's English is markedly different from current English, so too was Livius Andronicus's Latin different from, say, Ovid's. I think there was something like two-hundred-and-fifty or so years between their existences, so it might serve as a crude example to say, the differences we hear in the language of the early American nation and now, would have been something like the difference one would experience in "hopping" from 200BC Rome to 50AD Rome. Gratefully, we have extant writings from many periods of Roman history to compare and actually, see the differences in early period Latin and late period Latin, and, of course, all periods between.

The city changed an awful lot over time. Roman elite felt an obligation to improve the city often through big public works. That means no matter the time, Rome was changing. The biggest changes probably happened under Augustus. He built up the Campus Martius which was something like the Central Park of Rome, so that would be a noticeable new "district". The other big changes would be the theaters and baths. The first permanent theater in Rome was built in 55 BCE by Pompey, just over 20 years before the Battle of Actium. The Colosseum was built in 72 CE. Public baths existed in Rome in the first century BCE, but they really took off during the Principate. Agrippa built a deluxe complex on the Campus Martius around 20 BCE kicking off a trend of truly monumental structures. The Baths of Caracalla (completed 216 CE) were probably the largest structure in the world at the time.

Just as a small aside to the other points, I know that large scale fires were often an issue particularly in the poorer districts, so along with the large scale change of monuments, palaces etc. smaller neighbourhoods would be significantly altered also. A dark result of these fires is that enterprising young men like Marcus Crassus (who beat the slave revolt of Spartacus) could often make money by charging the owner of the burning building ridiculous quantities of money and simply not putting it out if the offer was not accepted. Source: Plutarch's Life of Crassus

Related, Suetonius famously said that Augustus could boast: "I found Rome built of bricks; I leave her clothed in marble."

Was this literally true? Did Augustus transform the physical makeup of the city of Rome that much? Would the time traveler from OP's question see that difference before and after Augustus?