VGA cable for 360 would work fine for you.
Also, an HDMI to DVI adapter should work, and may actually be cheaper if you have an HDMI model XBox. Unless the monitor is really old, HDCP for protected content (streamed movies, etc.) shouldn’t be an issue (early DVI displays sometimes won’t support HDCP). This should get a slightly sharper image at the native resolution of the screen, as it’s digital instead of analog. You’ll also need some way to hook up audio… as the monitor won’t have a pass-through (since DVI doesn’t carry audio), you would probably need an XBox audio adapter dongle thing (can’t remember what they call it, they sell it with their HDMI cable). If you don’t have one of those… well, it might not be the cheaper option anymore.
Either way, as for audio, if you’re planning on using computer speakers, you’ll probably need an 1/8" stereo headphone jack to dual-RCA jack Y-adapter. Couple bucks at RadioShack, cheaper online.