I do the lottery, rush and standing room all the time for Broadway shows. As a last resort I do 1/2 price a the TKTS line.
For a listing of all the rush policies, go to
www.allthatchat.com and click on "On the boards" to give you a listing of all the shows on (and off) Broadway. If you click on Show Rush policy, you will get a list of shows that have standing room, rush or lottery.
For standing room in NYC the show must be sold out. You can check box office capacity at
www.playbill.com the click link is on the left and is called weekly grosses. You can estimate whether the show will sell out or not. More sell-outs on the weekend than during the week. Sometimes you can ask at the box office if they are close to selling out, and come back later for tickets.
For a list of what's on the TKTS list go to
www.entertainment-link.com and click on Broadway (or maybe it is musicals) and then when you get the list, click on "what's on TKTS" for what was on TKTS for the past week. entertainment-link also gives you lots of info about the shows, too.
See if your university has a membership to Club Quarters. They have great weekend and holiday rates and three locations in Manhattan.