Are you SURE it does not have the red, blue, green, red white inputs? You should have a little plug that came with the tv and you plug it into that port which allows the red, blue, green, red white plug ins. Make sure you check your box to see if its in there.
It should look like this green plug.
If not then you have 2 options.
1. If your tv has VGA you could hook it up that way. The cable could cost around $39.99.
http://www.xbox.com/en-US/Xbox360/Accessories/CablesNetworking/Xbox360GVGSHDCable
2. Get a new 360. Seriously if you are using a 1st gen 360 I guarantee that thing will not last much longer. Especially if you start playing the new games that are coming out. I also have the 1st gen 360 with no HDMI and that thing has already been repaired or replaced 4 times by Microsoft for free.