Yes the "AD" numbering system was introduced by Dionysius Exiguus around 525. So he had to guess when year 1 was. In fact because Luke and Matthew give contradictory information, he literally couldn't get it right. According to Luke, Jesus was born in 6 AD during the census of Quirinius. According to Matthew, Jesus was born during the life of Herod the Great who died 10 years before the census in 4 BC. So Dionysius made a pretty good educated guess and more or less split the difference.

Prior to that the Romans primarily used named years, after the two consuls who held office in that year. So if someone were discussing the events of 59 BC, they wouldn't use a number, they'd say in the year of Caesar and Bibulus. Every now and again they would use how many years since the founding of Rome. Of course they didn't agree on when that had happened, so there was a reason why they didn't like that.

Most cultures numbered years according to similar systems, either counting from a significant event or from the beginning of the current king's reign. For example a lot of Greeks would count from the first Olympic Games. In general they would also include a reference to the system. They wouldn't just say in year 20, they'd say 20 years after the ascension of Philip.