Its possible, but there is a very real risk that you could end up with a trashed system.
You certainly want to plan on doing at least one practice run on a machine where you do have access to the console.
Its a good idea to get the serial console set up first and check you can control the bootloader using it. NB you'll still need something at the other end of the serial cable which you can connect to (might be a computer accessible via SSH, or a modem set to auto-answer).
Of course you'll need to know whether the IP address/name is hard-coded or obtained via DHCP, location of nameservers etc. Assuming its using grub as the boot loader - do read the grub documentation carefully and grab a copy of the curent grub.conf and menu.lst - you'll later need to merge this into the new setup before you reboot the system to ensure that you can get back to a known working version.
Do not attempt to install the new OS on top of the existing one! You should install it on a seperate partition. It's probably OK to use common partitions for /var, /tmp and /home but nothing else. That implies that you may have to resize the current filesystems.