I just found out that the 360 I've been rocking has no HDMI port whatsoever. I've been using Component all this time on my larger TV, but now that I'm moving out with a monitor that has no component input, I am kind of screwed. What do I do?

what about vga cables?

If nothing else, there are plenty of converters. Often cheap too. If you look on Google for "component to HDMI" in electronics you should find something that suits your needs.

Isn't there something that comes out of the 360 in whatever it is and it connects to the monitor in HDMI, kind of like the PS2 or PS3? I guess me asking this isn't really much help, sorry :(

You won't ever get HDMI quality then, but yeah: plenty of convertors. You say "monitor" though so you may have trouble splitting the sound and image.

Well you could always buy an arcade and whack your hard drive in, they run a lot cooler and quieter than the first gens

Lots of converters out there, but you want to pay a little bit more because from past experience I've learnt the cheap ones are flaky.

Congratulations on owning the only functional first gen 360 in the world. You should probably call Microsoft. Or maybe not, as they're likely to send a brigade of white-clad technicians over to your house to cryo-freeze your box and take it back to MS HQ for analysis.

Mine is actually still breathing too despite the fact that the optical drive is on the verge of a stroke and the sound it makes while reading something puts a Russian tank to shame. It's funny because that old crap outlasted my Elite which died yesterday while playing Alpha Protocol.
Back on the topic: you can buy VGA cables that are connected to the D-SUB socket of your monitor. I used to own one of those myself. The Xbox automatically works in the PC mode then which means 16:10 image aspect ratio instead of a 16:9.
Also, since D-SUB is an analog connection as opposed to the digital DVI, the maximum resolution possible for you will be 1366x768.