Wendy Wasserstein's Pulitzer Prize-winning drama will be directed by Pam MacKinnon.
Frasier cocreator David Lee will helm the Lerner and Loewe musical.
The celebration will take place on September 23.
The Lerner and Loewe musical with Broadway plans will play the Kennedy Center
Michael McElroy, Jennifer Leigh Warren, and others will also participate.
The theater was last home to the musicals Mamma Mia! and Rocky.
The play will feature a rotating cast including Brian Dennehy, Mia Farrow, Carol Burnett, and more.
Brooks is celebrating the 40th anniversary of his film Young Frankenstein, which was adapted for the Broadway stage in 2007.
The Armory will host two new productions from Trusty Sidekick Theater Company.