A night out in Stockholm on the cheap can be a difficult proposition, but it's not impossible.
To save on drinks, find the nearest Systembolaget, a government-owned liquor store which is the only retailer allowed to sell beverages with an alcohol content over 3.5 percent. The drinking age is 18 at bars, but you must be at least 20 to buy a Systembolaget. Be ready for long lines and short hours.
Make use of your purchases at the hostel or hotel before heading out around midnight, avoiding the ritzy club scene of Stureplan and Kungsgatan.
Concentrate instead on Sodermalm. Kvarnen (21+ on weeknights, 23+ on weekends) has reasonably priced drinks at both its H2O cocktail lounge and its beer hall.