Where are you seeing the popup and what is it telling you. That may help determine where the old email is and your possibilities on removing the pop up. For example, if an app was downloaded from the App Store with the previous Apple Account, an automatic update to that app will require the Apple Account used to download it. In that case, you would need to remove the app and download it again using your new Apple Account.
If you are seeing this email at > System Settings > [name on top] > Media & Purchases, that is the Apple Account used for purchases in the App Store and for purchasing media. If you need to remove the account from there, it may require resetting the Password if you do not know it, and you don't need access to the email account in order to use Account Recovery on the reset request.
Also if that is the account that is signed into your iCloud at > System Settings > [name on top], then Account Recovery can also be used if the normal reset options will not work for you: