Cool question. There was no organized system of numbering as in modern cities, though of course many streets and landmarks had names. Still, for a complete stranger, getting from the Forum to the "Fountain of the Four Turtles" could surely be daunting. Roman cities had vici ("neighborhoods" or "wards"). We know of dozens of named vici from Rome, and we hear about them in other places (like Puteoli) from inscriptions. At Pompeii, weirdly, we don't have the names of any neighborhoods, but we know they existed because vici and vicini (neighbors, denizens) are mentioned on inscriptions. Many vici seem to have been named according to the most obvious landmark in their area: a gatehouse, a temple, a public space, a fountain, an old statue, etc.

Speaking of gates: all of them had names in any city, as did the bridges, and most of the main thoroughfares. These would give immediate point of reference to the four points of the compass. Most ancient cities of any culture had a centralized area (forum, agora, market, etc) which would provide a more-or-less centralized point of reference.

There are plenty of inscriptions from the Roman world which reference streets which apparently had no name. Famously, an inscription at Cales (a town near Capua) says that an official repaired the road surface from "an alley next to the Temple of Iuno Licina all the way to the temple of Matuta." The roadway itself apparently had no name. There is also a famous one at Ostia which says that someone resurfaced an unnamed street running from the forum and connecting the two triumphal arches, obviously a main thoroughfare. Inscriptions like these throw a wrench in what we should probably assume: that nearly every object in the urban environment had a name of some sort, official or otherwise. Humans are inherently Namers.

If there were permanent street signs or other navigational indications in any Roman town, none survive, and not even the multitude of painted walls at Pompeii indicate any such practice. And so our stranger had to ask for directions. Most assume that direction-finding functioned like it used to function in more recent times (before GPS and smart phones): by increasingly narrowing the geography by a series of landmarks: "take the main road north from the mall until you see the fast-food place with the giant chicken out front; turn left at the chicken and then go through four stop signs until you see a tall wooden fence on your right; you are looking for the green gate with the stone Chinese lions flanking the front." No GPS or street names or house numbers required. In a Roman setting, perhaps: "from the river landing, follow the road from the Porta Iunonia until you reach the forum; from there, follow the lane to the right of the Capitolium north until you come to the Fountain of the Nereids; in that square is a little Inn called the Dead Gaul: look for a painted sign with a green shield; follow the lane to the left of that Inn across two side streets until you see (and smell) the bakery; the brothel is right next door to the bakery--and ask for Thracula."