Oooh! A favorite time period of mine. While I've never watched them specifically for stage managers, a couple movies from that time period that I can think of off the top of my head: All About Eve, Some Like It Hot (music, but lots of backstage antics), and possibly Singin In The Rain, though they're making a movie (but to me, it feels a lot like putting on a show). These are all early-mid-1950s movies, but I think that'd be your reference point. Have fun!