Hi @Avinash Jha ,
Check this document. As the article said:
it is clear the Address Rewrite Outbound agent will work only when the sender’s SMTP address is internal, and the session is authenticated.
Agree with the steps above all. You don’t need to sync mailboxes to exchange prem.
When the environment is set to hybrid, users will be considered as internal.
You could refer to this link discussed the similar issue as yours.
Microsoft Exchange Edge Address Rewrite with Exchange Online - risual
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