RAPP ARTS CENTER, 220 E. 4th Street b/w Avenues A & B. Many years ago I read a Village Voice article about this short-lived (late '80s to early '90s) institution--but I didn't save it for "the archives," and as such the fine details are lost to me. AFAIK, the RAPP (lord knows what the acronym stood for) was a multidisciplinary arts center, featuring avant-garde theater/performance art, dance, music, and visual arts. In addition to performance and gallery spaces, it also offered some living and studio quarters for artists--a college friend of mine knew a young writer who actually lived on the premises. The Center's 19th-century building had formerly been a Catholic school (all-boys IIRC), and the Catholic Archdiocese of New York still owned it--which caused some problems when the Church disapproved of the RAPP's presentations (either that, or they were just frustrated over non-payment of rent).
From time to time the RAPP would host rock shows. This current photo of the building's restored theater space looks a lot like how I remember it, but of course my memories may be faulty. Truth be told I can only recall being there twice--for a Redd Kross/Pussywillows/Embarrassment show, and for a big ska night featuring the Toasters, both in late '89. The only other rock references I've found online are to a Gwar/Mudhoney bill in November of 1988, and for a hardcore show at an unspecified date--but I'm sure there were others.
The building still stands and is now home to the Cornelia Connelly Center, a learning facility for girls from local low-income households. When not used for school productions, the auditorium space is rented out to various theater and dance companies, including the Metropolitan Playhouse.