I know they didn’t use the Gregorian Calendar system, but at 1 AD what did they call that year? What did they call following and preceding years?
A more comprehensive reply could be written, but for the general Roman year system, you might want to read a few links from the "How did the world agree on what year it is?" section of the FAQ. I think the best in this case are
"BC / AD - what did they use at the time?" by /u/azdac7 .
"Who or what group of people decided the switch from BC to AD, and why?" by /u/Algernon_Asimov (I mention this above others because it covers the topic and also the "Or..." link points to a longer explanation with grungy details). EDIT: further up in the chain was this , where he discusses an AUC calendar, but the link from the "Or..." mentions several.
This is not to discourage discussion. Further questions, data, and debate are welcome.