Well, Caesar was born in 100 BC and they knew how old he was. Simple calculation, really.
From the future perspective, yes. From the perspective of characters at the time, they wouldn't use that nomenclature, since it didn't exist yet. They would rather use the numbering system of their time (in this case, years since the establishment of Rome)
I had a thought once that no one ever lived in year one except retroactively (with some exceptions like the French Revolution). By the time a new system is adopted, it’s already been a while since the defining event.
Another possible exception: systems where year one happens repeatedly, e.g. Japanese era calendar scheme.
> with some exceptions like the French Revolution
It was also the case for the French Revolution. The republican calendar was established in year II. Nobody living in year I knew it at the time.
Likely for the same reason they speak modern French.