I think New World Stages has different policies per show (rush for Avenue Q/Jersey Boys, but student rush only for Puffs, and no rush for Desperate Measures which was there previously), so we probably won't know too much about Play That Goes Wrong until it's officially announced (though my guess would be that it will offer rush because the Broadway production did).
I think I read somewhere that you can only get rush tickets for Avenue Q in the first 20 minutes of the box office opening, but I haven't verified that, and I was able to get front row seats by showing up right when the box office opened. I think it was a Saturday and we were the only ones there at 10 am, though I'm guessing there might be a higher demand now that they've announced closing. I rushed Jersey Boys on a weekday this summer and was maybe tenth in line at noon when the box office opened, and we got three tickets in the back row of the orchestra.