Just because the newer systems have CPU's with built-in GPU's does not equate to your system, Sorry ;-{
No, you can't run your system without a graphics card.
It might be a simple problem were the thermal paste is not working and/or the fans are not cooling the system properly. Are you up to fixing your system?
In any case you should be able to migrate your data off of it. If you are thinking of getting a new system. You can try setting up your system in Target Disk Mode, follow this Apple TN: [http://support.apple.com/kb/PH13842|Transfer files between two computers using target disk mode]. Or you can remove the disk drive and using a SATA to USB adapter connect your removed HD directly to your new Mac.
You may want to consider getting a used system as most of the newer models are much harder to fix. And besides a used unit will be cheaper.