OAuth Updates for an App for just One Single User (Developer is only user) required ?

In your case, you're not actually processing the OAuth app authorization flow, since you're just using a single pre-generated long-lived access token, so you technically don't need to change anything in your code.

While long-lived access tokens are now considered deprecated, but we don't currently have a plan to disable existing long-lived access tokens. (If that changes, we will of course announce that ahead of time.) That being the case, you can continue using existing long-lived access token(s) like this. Note that after the change you won't be able to create new long-lived access tokens though. So, after that point if you revoke your existing long-lived access token you wouldn't be able to generate a new one, and would need to instead update your code to use the new short-lived access tokens/refresh token functionality.