Usually, just starting the game up while you are connected to the PSN after you renew will update the expiry date. If not, go to the store (or your download list), start the download, wait for a few seconds, then cancel it. THEN start the game up, and it should update the expiry date.
The license file for a free Playstation Plus game will not be renewed until it expires, because it doesn't need to be. Renewing your Plus subscription does not change the expiration date for any content that is already installed. It only changes the expiration date for content that you download in the future. The expiration date of the content is not the place to find out when you Plus subscription expires. You need to do that in the Account Management app. Just ignore the expiration date of the content. As long as you can start the game, you don't need to worry about it. It isn't necessarily supposed to be the same as when your current subscription ends.