Hello, @StefanoB,
Great to know that the issue has already been resolved and thanks for sharing the solution so that others experiencing the same thing can easily reference this!
Since the Microsoft Q&A community has a policy that "The question author cannot accept their own answer. They can only accept answers by others(https://docs.microsoft.com/en-us/answers/support/accepted-answers#why-only-one-accepted-answer)", I'll repost your solution in case you'd like to "Accept (https://learn.microsoft.com/en-us/answers/support/accept-answer#accepted-answer-in-a-question-thread)" the answer).
Issue Symptom:
Whether there is a way to sync admin exchange on cloud with on premise?
Whether the process can be focused?
Resolution:
The author added in exchange console the user with this command:
enable-RemoteMailbox -Identity "name.surname" -RemoteRoutingAddress "******@mail.com"
Hope you can consider accepting the answer as it could help other members of the Microsoft Q&A community who have similar questions and are looking for solutions.
Thank you for helping to improve Microsoft Q&A!
Best Wishes,
Alex Zhang