My question is what kind of paperwork/registration do I need to complete in order to legally own this?
You don't need to do anything but pay the man and go home. Arizona does not have any kind of laws that require you to register any kind of firearm or any paperwork that needs to be filled out when two provate individuals conduct a firearms sale.
Having said this, depending on how good of a "buddy" this guy is, you might want to get a signed bill of sale for your records just in case he isn't as good of a buddy as you thought and reports the gun stolen or something.