Yes I heard Papparazzi but never been there, I will check for sure but as I remember its not that inexpensive?
It's not especially cheap and the opposite of cozy.
Try some of the bars in Chinatown (the pavilions behand Nowy Swiat 20). Lot's of little bars, none of them expensive, most cozy, some quite trendy.
Or, if you're students, there's Pub Nora (the burrow) - a Warsaw students' place for years, off Krakowskie Przedmiescie. As for Bar Warszawa, the front place and the one at the back are in effect different bars. the upstairs room at the back feels like an old-fashioned living room, and it is quite cheap.