The free event will take place at Joe's Pub.
Ivey and Houdyshell will jointly participate in the "Acting Aloud" series.
The production is produced in association with London's Royal Court Theatre.
The 1960 musical completes the company's Musicals in Mufti Series celebrating the work of lyricist Sheldon Harnick.
Urie will depart the off-Broadway production in March to embark on a national tour.
Williams will join the cast this April.
Directed by Tony winner Doug Hughes, the production will also feature Kristen Bush, Barbara Garrick, and Kevin O'Rourke.
The original West End production of the 2011 Tony Award-winning Best Play will be broadcast across the country in a one-night-only event.
Christopher Durang's 2013 Tony Award-winning Best Play is a Chekhovian hit out west.