Without knowing the OS of both systems, I wouldn't be able to give you specific help about transferring files via cable, but here's some general ideas:
In my opinion what you should try to do is make a backup of what you have on the hard drive of the computer that is malfunctioning while you still have a chance to do so. You shouldn't need to make a complete drive image if you still have all the programs you installed (as well as their serial #'s), you'd only need to backup your work product. Of course if you don't have copies of your programs or their serial #'s then a complete backup would be in order. Once you have your data backed up then you can go about fixing your problem.
If you have a CD burner installed in this computer, making a backup is easy. If you don't have a burner in this computer but have one in another computer you own or have access to then pull the hard drive and burn your backup.
If you don't own or have access to a CD burner you should think about pulling this hard drive and making a backup to store on the hard drive of the other computer, assuming you have the space to do so. That would be quicker than transferring over a cable.
Once that's done then you can try and figure out why your computer is shutting down every hour or so.