Hi @prabhash-jena ,
1.For the room mailbox. If you don't need to create a remote mailbox or manage it through On-premises Exchange management tools. You can create it directly in Exchange online. If needed, you could create it on-premises Exchange.
2.For the shared mailbox, I recommend that you first create Shared mailbox in on-premises Exchange server. Because of this known issue I found below, if we create a shared mailbox directly in Exchange online, it will may cause the following problems.
Please refer to: Users in a hybrid deployment can't access a shared mailbox that was created in Exchange Online
1).Users can't open the shared mailbox in Outlook.
2).Users can't view free/busy information for the shared mailbox.
3).Users can't send mail to the shared mailbox.
About remote mailbox, please refer to:Enable-RemoteMailbox
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