The play is currently in previews at The Public Theater in New York City, opening March 15.
The Broadway musical's final performance will come just before the production tours North America.
The "SNL" alums will reprise their roles from the original film.
Once Upon a One More Time will open this spring on the Great White Way.
Rice is the first songwriter primarily known for his work in theater to receive this award since Stephen Sondheim in 1999.
"At the end of the day, what matters the most is how you make people feel," the Funny Girl star said.
"Speaker of the House/ Cunning little chap/ Covetous, conservative and full of crap," Rainbow sings of the embattled GOP leader.
Richard Linklater is directing the movie, which also stars Beanie Feldstein & Ben Platt and will be shot over two decades.
The Disney Channel icon shared photos with the Glee alum after seeing the Broadway show.
The Broadway star blamed "drunk scrolling" for her hysterical gaffe.
