Q: Problems with app store

The apps you are trying to update must have been purchased with the prior owners account. You would need to delete his apps and buy your own.  You could then update them using your account.

Be certain you are starting with a clean slate on your iPhone.  It sounds like you have not done this.

The iPhone should look software-wise as if it's a brand new iPhone coming out of the box for the first time before you take it over.  This explains how to do that.  Once done, you will never be bothered with any "previous owner" requests again.

What to do before selling or giving away your iPhone, iPad, or iPod touch - Apple Support

What you are experiencing is the result of the old owners apps still being on the iPhone. iTunes will always ask for the password for the Apple ID that was use to download an app even if the device is signed in with a different Apple ID. You can either delete the apps not associated with your Apple ID or erase the device and start over.