iPad 1 Software Update
Yes, you can use your Mac and not trash anything. (Especially if you know how iTunes syncs to iOS). 

For simplicity's sake, you might want to create a new account on your Mac so your friend can enter her Apple ID account in to the Mac if needed to update apps.

This also keeps her backup separate from your backup and simplifies life if you want to transfer her purchases to iTunes (which iTunes will want to do to ensure it can restore the iPad should the update/upgrade fail for some odd reason).

I would say, just start with getting a good backup to a new account and then see if the device even needs an update. If you connect her device to your iTunes on your account - you might have some learning and clean up if her contacts mix with your contacts, etc...