Yes your monitor is HDCP (HD compatible). I own a sweet 22" LG widescreen LCD and I love it, a top of the line TFT, but yours isn't bad either and you shouldn't have any problems.
Although keep in mind that that with this and all monitors in this class that are HDCP that you will only be capable of 720p. But it looks great, some even say 720p can be better than 1080i anyhow.
Anyhow, I own a PS3, but as far as this topic goes, same difference. Not too familiar with the 360, but of course I assume it has an HDMI input.
Then again, I did hear something that the 360 didn't have HDMI ports until those manufactured this past July. Really? I'm a playstation guy, but I would of been pissed to get a 360 with no HDMI output, even if you don't need it now, you will in the future.
Anyhow, if you want to hook up to your monitor and you do have an HDMI out port on your 360, then you want a simple HDMI-DVI cable. Under any circumstances you do NOT want the product you linked to, that cable is only for TV's, not monitors. You can see just by looking at it that it has no plug for your DVI, it just has HDMI on both ends for an HDTV, then that other part is for audio jacks I guess.
You can get an HDMI-DVI cable probably for as cheap as $15 for a generic one on newegg or something. I tried a couple more expensive brands, and I must suggest getting the RCA HDMI-DVI, which you can get probably at your local walmart for $28 like I did. You can try cheaper ones, but I first tried a Phillips that was just a tad less expensive than RCA and it sucked ass, don't buy that one. Get a good one, it is an important cable. Nice thing for me since I have only one DVI port is that I can switch my normal DVI cable (from my gpu) with the HDMI-DVI cable in just seconds, this RCA one works without even needing to screw the sides in.
Now you got sweet 720p video. Personally I'm not rich and never owned an HDTV and will be a while until I do. And finally being able to play a console in HD at 720p is just a world of difference. Was hooked up to my 26" flat screen normal TV, but due to the jaggedy look of non-HD, the graphics didn't look a whole lot better than the xbox or ps2. But the 720p, although only on a 22" widescreen monitor, made a world of difference. Amazing!
Anyhow, once you get your video hooked up, then you need to get your sound hooked up through your sound card or onboard sound. That is a whole other topic, won't get into that unless you ask, there is probably info on that all over the place.
But there is a few different relatively simple and cheap or free ways to do this, even if you sound card is used up or you don't have the right cables/adapters/converters, you can still run sound through your sound card, although not directly. 720p plus ps3 sound running through my X-fi Xtremegamer is just awesome!
I would elaborate but I'm not sure what you are working with there.