If Linux does see it i don't see why you couldn't back up your files. It depends on whether or not the hard drive is spinning and working. I have had drives that show up in disk management but the drive itself has seized up. It might be a similar issue to yours.
First i would suggest that you verify that all cables are hooked up securely then when you boot your PC up, listen to see whether the drive sounds like it's spinning up. If it is making a strange or loud noise then you have a problem. If the drive seized up it's possible you might hear a buzzing noise at least this is what one of my drives did when it died. Anyway if Linux does see the drive, make sure you have the back up drive ready!
I'm not too familiar with Linux drive readability but it can't hurt to try it out since it's free.