This is another funny achievement to earn in Bioshock Infinite. To unlock Lost Weekend, you need to have consumed 3 alcohols while engaged in combat with enemies. To do this, you'll need to know where the stash of sky-high booze is before you shoot the people shooting at you.
You'll know you're good and proper trashed when your vision starts to blur. Whip the shotgun out and blast away. If you're having trouble locking an enemy down in your sights, use Murder of Crows to hold them in place. Powerpyx.com has a youtube video so you can see how it's done: