Hello,
If you see the 'Old email address' to verify your identity, you can replace it here by signing in to your account on the Security Settings page of your Microsoft account.
https://account.live.com/proofs/Manage
However, you need another verification method to sign in to the security settings page.
If you signed in, add a new way to sign in to your account first.
Then remove the old email address used for verification.
If you no longer have any other methods to verify your identity, you will have to replace the Security Information.
Select "I don't have any of these' and follow the prompts to replace the Security information.
Note - If you remove all of your previous security information and replace it with something new, your account will be put into a restricted state for 30 days.