Old, but the standardization we have today is not as old. Many ancient Greek cities as well as the Romans would keep track of time by tracking the magistrates that served that year. The Romans would, for example, use the list of consuls, who each served for 1 year, to keep track of time. This could get very complicated, however, see this example from Thucydides History of the Peloponnesian War
For fourteen years the thirty years' peace which was concluded after the recovery of Euboea remained unbroken. But in the fifteenth year, when Chrysis the high-priestess of Argos was in the forty-eighth year of her priesthood, Aenesias being Ephor at Sparta, and at Athens Pythodorus having two months of his archonship to run, in the sixth month after the engagement at Potidaea and at the beginning of spring
What's going on here is that Thucydides wants to target as wide an audience as possible. He wants people outside of his native Athens to be able to understand his history, but he can't do what a modern historian would do and simply number the years, because there is no standardized year number for anyone, much less a shared one across Greece. So what he does orient to the magistrates and priests of several major cities and major events, in the hopes that everyone reading his work will know when at least one of the events happened, allowing them to place the rest of the events in the history relative to the event they know about. So as you can see, dating events for a wide audience was extremely difficult, and required some creativity on the part of a historian to actually place events in time for everyone reading.
So sixteenth century specialist here. In general any one involved with the church or state would certainly know the year as it would be cited anytime a formal document was written. This would include officials, professionals, clerics, and probably many tradesmen.
Now, plebian people might not have as good an idea. Having read many criminal cases from the sixteenth century most everyday people didn't reckon the passage of time using the calendar year. Most people who were not in the habit of keeping track of the specific date tended to remember the past through association. So they would have been more likely to remember that something happened the year that "so and so was executed" or "so and so became Bishop".
Similarly recounting the passage of time there tended to be more variance between individuals. Witnesses to a wedding could vary in their recollection of how long ago it was sometimes by several years . One person might remember it as being 5 years before while another might claim 7 and a third might say 10, even if all three agreed on the details.
This brings up another interesting question: when was it decided that a year would be 365 days long?
Edit: don't really understand the downvotes. I meant to ask when people realized that a year was 365 days long. But I could've worded my question better.
When did our current year count get standardized?
Could anyone comment on Christendom at the turn of the first Millenium and fears or anxiety among people in Western Christendom regarding the coming of the Millenium and the belief that it would be accompanied by the Christian Apocalypse? I have heard that this belief was widespread at the time, and it would certainly seem relevant to OP's question as to whether people were aware of the date in their day to day lives, but I am unable to find more information regarding the historicity of the phenomena on my own.
FYI, there have been previous responses on this topic that may be of interest