Certainly most of the Roman citizens knew about the myth of foundation of Rome and other myths of heroes, but did they know about their own actual history? Did the people who saw the last emperors know the names of the first ones and the years in which they ruled? Did Romans know of their history or did it take until historians from our times pieced everything together to get a global portrait?
The city itself was a living monument to figures of the past. There were dozens of 'victory' temples put up by conquering generals in the middle Republic that commemorated their name (and their gens name) which would be a physical reminder of their victory over the 'X' in the 'Y' war. Then you have various victory arches or other imperial monuments that graphically depict the achievement (e.g. Trajan's Column, Arch of Titus). You also had various other public monuments or buildings depicting other events from Rome's history (e.g. the Temple of Castor and Pollux where they were said to have appeared after the Battle of Regillus to announce Rome's victory).
There was a list of all of the consuls of Rome that was freely visible (although their original location is debated). This ran from the early Republic to 12 CE. I would presume that there would be other public lists of the consuls in imperial times. So if someone said something happened in the year Fulvius Paetinus Nobilior and Aemilius Paullus were consuls, you could look that up and work back that it happened in 255 BCE.
When you get into the imperial era, the emperors (or their close associates) tended to monopolize massive public buildings in Rome. These were clearly associated with their building (e.g. Forum of Trajan, Baths of Hadrian). I'm not certain if all did, but some displayed their builder (e.g. the Pantheon states it was built by M. Agrippa who was consul three times) - and one could then look up that name in the consular lists to figure out where they stood. This of course brings up the question of literacy (or even functional literacy) which is another argument.
So there's a lot of "big man" memory aids scattered throughout the city.
Being that many of our sources are the existing writings on this very subject from contemporary historians such as Livy or Suetonius, I'd wager that yes the Roman public did understand their own history. That said, was it mainly the educated elite classes that had this knowledge or was it wider spread in the plebs? That is a different question that I couldn't answer with any sources.
Follow up question: do we know if any of the primary sources we use were used in formal Roman education? That is, would your tutor assign you Tacitus?