iPod classic hard drive upgrade
Yes, you can replace the hard drive in an iPod Classic. There is no need to replicate anything on to the drive as all the OS is stored in firmware separate to the drive. Once there is a blank drive in the iPod the firmware will take over. Instructions are here at iFixIt.

Mounting the Classic to take a backup is another question entirely. Since you can't actually get to the OS to back it up the only thing you need to backup is the music and I assume you have a copy of that already. If you don't then I recommend Senuti to recover a copy of your music and playlists to your Mac.