Translation:Cheers! Happy New Year!
I think that "bottoms up" should also be accepted. I also think that "bottoms up" is closer in meaning to "干杯".
While "bottoms up" is in the dictionary, 干杯 also means to "propose a toast." People tend to clink glasses and say "Cheers!" more often than they say "bottoms up", which is informal and used mainly by men or in bars. It's possible that someone has said it before, but I've never heard "Bottoms Up! Happy New Year!" used together as a phrase.