Translation:Cheers! Happy Birthday!
干杯 (cheers) is literally "dry cup" and of course referring to a drink (toast) not "cheers" as a casual greeting.
Actually they really are the same, in the original Traditional Chinese characters and in their meaning. 乾杯!
干杯 literally means "to empty the cup", so the correct translation should be "bottom up" or something that conveys the idea of having the cup empty. "Cheers" is too superficial a translation.