at the very beginning of the New Year; early in the New Year
Often used when something happens right after New Year's Day, sometimes with a nuance of 'already' or 'so soon'.
Right at the start of the New Year, work got busy.
I'm sorry to bother you so early in the New Year, but please contact me as soon as possible.
Compound of 新年 (new year) and 早々 (early, promptly). The exact historical derivation is uncertain, but the combination naturally expresses 'at the very beginning of the new year'.