For one thing, INSERT and UPDATE do to different things: INSERT inserts new records and UPDATE updates existing records. There's unlikely to be many cases where you'd be uncertain about which to use.
For your problem, I'm making guesses about your database structure and the sort of data you're storing, but if item_name is the name of an item then I guess it's not just a number; so you'd want to be putting it in quotes in the query - just as you are already doing for username.
UPDATE users SET inventory='$item_name1' WHERE username='$auth_user'
If I'm right, what would have been happening would have been that you're sending this broken query to MySQL, MySQL is choking on it and sending back an error message, and you're not making any attempt in the script to find out whether the query succeeds or not or what the error message is if it doesn't.