Q: Can I use a back drive that I had on my PC on my new Macbook Air? Or do I need to purchase a new backup drive?

You should be able to use a drive that was previously used with a PC. You will likely need to reformat it. Note the following quote from Mac Basics: Time Machine backs up your Mac - Apple Support

Your Time Machine drive needs to be formatted as Mac OS X Extended (Journaled). If you select an NTFS or FAT-formatted drive, Time Machine prompts you to reformat the drive. Important: Reformatting erases any files on the destination drive. If you're not sure if you want to erase the drive you connected, choose a different drive.

The following gives instructions on how to reformat to Mac OS X Extended. It refers to Aperture but the instructions would be the same: Format external drives to Mac OS Extended before using with Aperture - Apple Support