Auld Lang Syne is well known in Russia – and throughout Russophonia – both as a melody and in Samuil Marshak's translation of Robert Burns' poem, though not necessarily as a New Year celebratory song. The Russian version is called Zastolnaya – Drinking Song. Can't limit this just to New Year, can we?
Забыть ли старую любовь
И не грустить о ней? Забыть ли старую любовь
И дружбу прежних дней?
(Full text is here)
This is the 1973 song version by the Belorussian group the Pesnyary. The sound is not very good, but it gives an idea of their trademark folk-pop interpretation and a distinct Belorussian accent.
Music by O.Yanchenko, original lyrics by Robert Burns, translated by Samuil Marshak. Leonid Bortkevich, the group's lead singer, was married to Olga Korbut, the legendary Soviet gymnast, also a Belorussian. The photos in the clip are of them as a family couple.
And here is a humorous medley, that includes the Pesnyary version too: