Anita Balan said: "Hello, those of us who are not in NYC, (travelling from DC, Philly or Boston etc...) for a weekday or weekend show... What are some of the shows for which we can try Rush can get decent tickets? What are the best ways to try Rush for popular shows like Moulin Rouge, Hamilton or Hadestown? Thanks in advance."
http://www.playbill.com/article/broadway-rush-lottery-and-standing-room-only-policies-com-116003
Hadestown and Moulin Rouge have standing room, but otherwise, you have to enter the online lottery.
Hamilton, your best bet is the cancellation line for regular priced tickets. There is a pretty detailed thread if you search on here.
Depending on what time of year you are coming in, you can always check stubhub.
I'll put in a plug for Come from Away if you haven't seen it. Rush/standing room tickets are excellent and if you are in standing room, its one of the better theaters for it with excellent viewing angles and at a 100 minute show without intermission, standing is not bad.