There are no local specialties as far as drink goes, which means that the usual collection of water, juices, tea, coffee and softdrinks are readily available.
Alcohol is mostly only available within top-quality hotels. Cheaper alternatives include the Fujairah Marine Club on the corniche and Fujairah Beach Motel, which also has a retail "hole-in-the-wall." You'll have to ask as it's hidden away a bit.
The best value drinks in town are to be had in the pub at the Tennis and Country Club go on a Tuesday and you get karaoke too.