I also was last at Yankee Stadium in 2008 at the old ballpark. I've not been to the new one yet so I can't help much on that. I know the new place is "different", but I'd still get there a little early so you can explore the park, go check out the monuments, etc. That stuff definitely got busy and crowded in the old park, not sure about the new one.
There's some good sports bars around the area. Stan's and some others.
Taking the subway to the game is definitely cool...the energy will start to build as more and more fans in Yankees gear fill up the cars. Beats the hell out of twiddling your thumbs sitting in traffic (and is a lot cheaper!) If you ever go to a game in D.C., take the subway down to the stadium, too.
As far as where to stay...Midtown is certainly a good, well, "middle" area that won't make it hard to get anywhere. I'd check out some food options in some of the ethnic enclaves on top of some of the fancy NYC dining options (Chinatown downtown, etc. If you're feeling moved, Little Ukraine is in the East Village area. A place called Veselka is good stuff without breaking the bank.)