Component Out is analog. Its capable of 1080p output. The original Xbox 360 can only output analog.
So the question is, what sort of other connections can your Xbox output at 1080p?
Answer:
Amazon.com: Xbox 360 VGA HD AV Cable: Video Games
VGA output is capable of very high resolution output. I'm currently using it to display 2048x1536 @ 75hz on my own computer. The Xbox VGA adapter should be cable of 1080p, just the same as component output. What you would need to check is if your Monitor supports analog input. Judging by the specs you gave me, the answer is no.
There are multiple types of DVI connectors that I don't feel like looking up now. DVI-D is a digital only connection. DVI-I, I think, supports both analog and (dual link) digital. If your monitor supports DVI-A/DVI-I, you can grab the Xbox 360 VGA adapter linked above, then get a VGA to DVI-A adapter.
.... Also for anyone else reading this. The eForcity VGA to RCA/S-Video adapter will not work in many setups. VGA is electrically different from RCA/S-video, therefore needs an active adapter. Not passive as the one linked above. That cable is designed for Video cards that have a setting to either use the VGA port as an actual VGA (D-Sub) port, or an RCA port.
D-Sub to DVI-I adapters work because the signals are electrically the same. DVI ports were designed to carry both the D-sub analog signal and digital DVI signal. You just have to double check if the receiving and sending devices are support the same signal type (analog or digital). In the case of your Xbox 360, its outputting analog, therefore you need to check if your monitor can receive analog.