Using the old computer with the new monitor, are you able to enter the Advanced Options menu? Power on the computer an tap F8 until you see the Advanced Options menu........If you are successful you will see options such as Safe Mode, VGA mode...etc Are you able to enter this screen?
If you are able to enter this screen, try VGA mode... You won't get the best resolution/color, but you will at least be able to install necessary drivers for your system..
edit:Sorry, you won't be able to get online to get updates obviously, until your net. card driver is installed.. Is your network adapter onboard or is it a card? If you can figure it out, what is the make/model of the network adapter? What is the make/model of the computer as well? Is there another disk that came with the computer when you bought it? Some computers come with separate disks that carry the drivers...
I did an install of a Dell system once, and had to get the network adapter driver off of a separate CD, that came with the computer...