The problem isn't that he was the "first" user (this ain't Windows) or has the ability to administer the computer.
It has to do with the AppleID that was used to purchase the software you are trying to update. He bought it and only he can update it.
This is what he should have done before selling his Mac:
What to do before selling or giving away your Mac - Apple Support
AppleID's and their purchases cannot be transferred, merged, or anything of the like. You need to delete the software you are trying to update and purchase it yourself. That way, it will be your AppleID that is used to update it.
However, as you have no idea what the seller could have installed on the Mac you purchased, you should completely erase the Mac as per the linked article and then purchase any software you need. Backup your Mac first, so you can use the Migration Assistant to Migrate your user back to the newly installed OS, but don't Migrate any Applications. You'll need to get the OS updated back to the current version you have before some things will work correctly with the new data structures, so when the Startup Assistant runs, create a user that doesn't have the same name as your old user. That will make migration of your old user account much easier.