If you don't drink, that's your business, and you don't need to be ashamed to just ask for a straight glass of soda or juice. Any decent bar should at least stock Sprite, Coke, soda water, tonic, orange juice, cranberry juice, pineapple juice, and possibly grapefruit juice. So basically any of those or any combination of those is what you should expect to choose from.
I'd also advise that if you plan on hanging out for a while or becoming a regular at any bar, it is definitely a good idea to visibly tip well on your drink orders, and perhaps tip on each refill you get. Not drinking alchohol is not a big deal at all, but there's a big difference between being known as "that nice guy who likes hanging out and meeting people and having a glass of soda since he can't drink" and "that cheapskate who sits alone for hours sucking down free refills and trolling for girls while taking up space that could be occupied by higher spending customers."