Xbox 360 is HD, which means 1080i
Actually the native resolution of most games on the console is 720p (which is also HD). A few games can do 1080p, but most cannot.


Re original post: you can set higher resolutions with the monitor though whether this causes the game to display in true HD will vary. Some games (I believe Gears of War is an example) will scale the output so it displays a letterboxed 720p image.

In theory the monitor should look better. And yes it will fit right if you select the right resolution as the output.