Old Apple ID

If you are simply usinga different email address with with no change in Apple Account then you don't have to do anything different. It will recognize the apps are still associated with that particular account, not an email address as such.

If you changed the address and not the account and you are now seeing an old email address then you have something else happening on your account. What I do not know because I can't think of an easy way to accomplish that. It may be you do indeed have a second Apple Account set up somehwo and the apps are assoicated with that one and not your primary one logged into the device. You can go to your Apple ID, then Media & Purchases and see if you have a different account set for those.

By old do you mean you changed the email on one account, or do you mean you had two accounts? If the latter then realize that items purchased with one account cannot be updated using a different account on a device. You have to delete the apps and download them with the new account. If this is the case then you can look into using Family Sharing to share them with yourself.