This is likely entirely too late for your birthday, but on the off chance that you're still poor and interested in economical boozing on the Hill:
(1) The Redwood is your friend. Insofar as booze goes. The happy hour was just shortened, but still runs from 4 to 7. The $3 wells are about as cheap as they get on the Hill (though, if you're looking for an early funk for less, try the happy hour at the Broadway Grill). The real deal is the $4 beer/shot combo, and $6 pitchers. And swill beer is always $2/pint, $1.50 can, less during happy hour. Pours vary, depending on your barkeep and how much you tip, but are generally solid. Food is normal prices, but the burgers are solid. And the tortilla chips are fresh made and beautiful.
(2) I second the StrangerMart recommendation. But be advised that those coupons are delivered by mail (or were the last time I used them), and so you need to plan a couple days in advance. Captain Black's happy hour is meh booze-wise, but the food menu is the shit. $3 deep-fried mac-n-cheese? Yes.
(3) The Monk. Thursday nights. People tend to forget about their long running cask night. 3.00 or 3.50 buys you an imperial pint of generally severely potent cask aged microbrew. They don't sell this shit in pitchers. For good reason. Don't forget to try the doorbell, but you really have to hammer on it these days.
(4) If you or any of your friends has an iOS device, get the Cocktail Compass app. It's not comprehensive (you won't find the Redwood or the Monk on there), but if you're happy hour hopping on a budget, it'll help you bridge the gaps between late night happy hours and afternoon ones. And remind you of useful all day ones. For example: The Cha Cha, while generally above average price-wise, has a happy hour all day on Sunday and Monday.
Happy drinkening.