It depends on who's talking.

The only people using that month/day calendar system at the time were the Romans. They didn't start to use year-numbers for another three hundred-odd years. So they called it "pridie Kalendas Ianuarias consulibus A. Plautio et A. Caecina Severo" (the day before the Kalends of January, in the consulship of A. Plautius and A. Caecina Severus).

(This is neglecting the fact that the Roman calendar was a little bit screwed up because of a misimplementation of the Julian calendar. It wasn't fully corrected until 8 CE, so it's a few days off. We don't know for sure the details of how the correction was implemented: there's not much point trying to guess the precise day.)

Modern ancient historians use the Roman system for day-number and month-names. And they use the year-count that was adopted by early mediaeval historians like Bede. So for them it would be called 31 December 1 BCE (or BC).

In some disciplines, like historical astronomy, it is the convention to ignore CE/BCE and AD/BC, and treat year numbers simply as integers, and use the Julian calendar for all dates prior to 1582. So the date you specified is simply in year 1, not 1 AD. 1 BCE gets called year 0, 2 BCE gets called year -1, and so on.

And there are a few other notations floating around. For the purpose of carbon-dating, year 1 would be 1949 BP (Before Present), and the year before would be 1950 BP. Not much point in that, though, since carbon-dating isn't precise enough to treat it as that kind of calendar).

And at the time, many other non-Roman calendars existed in other parts of the world (notably the Seleucid, Han, and Alexandrian calendars): I omit those because "1 AD" didn't exist in those calendars.

To add just one more datum to /y/Zikzax's excellent answer: In the areas of China controlled by the Han dynasty, that day would have been some date in the 11th or 12th month of the 2nd year of the Yuanshou Era. (One could calculate the exact date on the Chinese lunar calendar by first determining what date on the Western calendar was the first day of that Chinese year.)