It doesn't matter if the email address exists or not. That the is the Apple ID under which the apps were downloaded. You can sign out of your current Apple ID and sing in under the old Apple ID to update the apps, which is a pain. Or, since apps are forever tied to the Apple ID they were downloaded under, delete the apps and redownload/repurchase them under the current Apple ID. There is usually no good reason one needs to change the Apple ID. One can change the email address associated with an Apple ID, which is what you should have done, but doing that now will create even more issues for you.
If I were you, I'd delete the apps, rebuy then under the Apple ID you use and remember that you don't have to create a new Apple ID even if you get a new email address as that can be changed.