Around the time of the 2nd Punic War the population of the city was relatively small still, hardly a cosmopolitan, sprawling metropolis. For much of the population, directions wouldn't have been much of a problem until some time later.

Some streets, like the Via Sacra, had names, but Romans navigated by landmarks. The city was filled with landmarks, some large and obvious (the Portae, the Campus Martius, the forum, various public buildings, etc) and others more locally known (statues, people's houses, that road that leads over there).

Duncan's comment about the city's streets being "the secret shame of Europe" is both editorializing and anachronistic. For one thing, Europe didn't exist yet. For another, even if we're restricting this to the shame of the Roman state, I don't see any reason why it should be so. Plenty of other cities were sprawling, disorganized messes and at the time of the Punic Wars Rome had hardly founded any large colonies at all--the organized grids of Roman foundations would not become common for some time yet. Even during the late Republic and the Principate most large cities were poorly planned, and the neat grids associated by so many with the Romans were generally restricted to veteran colonies, most of them in the provinces--and not because of any program of urban planning, but simply because most of these colonies were built up from fortified legionary encampments. I've never seen anything in any ancient text that would support the ides that people felt ashamed at the streets of the city at all, at least nothing important enough to remember