I use my xbox and my pc on the same monitor, DVI for pc VGA for xbox. Got a samsung 2443bwx 24 inch widescreen

.
For the price it is an absolutely kick ass monitor, this is what I use
http://www.newegg.com/Product/Product.a ... 6824001295 The native res is 1920x1200, but I play at 1920x1080 (1080p) on my xbox and it works great. I then use the fiber optic sound output on the vga cable to go to my 5.1 sound receiver for full surround sound and badaboom a very nice setup in full HD with kickass sound, without a TV lol. ( I also use my monitor with my HD cable box for 1080p TV too) so I actually don't use or need a tv.
The xbox will practically support any monitor with a VGA input, the xbox supports most res's beyond 1024x768, it even supports 1680x1050 now since the new update. Any monitor with a native 1920x1200 res should be able to support full 1080p as well.
The monitor link you posted is compatible because it has a VGA input and the xbox supports 1440x900 res so u would be just fine with that monitor. If you can though I suggest a 22 inch or 24 inch wide screen monitor, because these are really good sizes to replace a tv (sorta a sweet spot):D Hope I helped
Also if your pc is gonna be next to your xbox and u have a sound card u can plug the 360's sound into the "line in" on your pc and have it come out your pc speakers. You can do this because with the VGA cable they give u an adapter which turns the red and white RCA plugs into one mini plug which can fit into your pc's soundcard