This is one of the most requested features in an Exchange hybrid scenario (i.e. Exchange Online combined with Exchange on-premises) and as of early February 2016 it is finally officially supported: Cross premises Full Access Permissions.

This means that if you have a manager’s Mailbox on-premises, and an assistant Mailbox in Exchange Online, the assistant can open the manager’s Mailbox. This works both ways, so if the manager’s Mailbox is in Exchange Online and the assistant’s Mailbox is in Exchange on-premises the results are the same.

There are some caveats however:

• This only works when Full Access permissions are granted, and this is achieved using the Exchange Admin Center or Exchange Management Shell in Exchange Online.
• Send-As, Receive-As and Send-on-behalf-of permissions are not supported cross-premises.
• Your Outlook 2013 should be patched with at least the November 2015 update.
• The first time users open a Mailbox in the other organization they might see a credentials pop-up

The people picker in in the EAC in Exchange Online supports adding Mail-Enabled Users (MEU) and regular Mailboxes, so you can use EAC in Exchange Online to add cross-premises permissions. The EAC in Exchange 2013/2016 on-premises only supports adding Mailboxes, so the online version of EAC need to be used.

More information can be found on the following Microsoft articles:

• Permissions in Exchange hybrid deployments – https://technet.microsoft.com/en-us/library/jj906433(v=exchg.150).aspx
• https://technet.microsoft.com/en-us/library/jj200581(v=exchg.150).aspx#considerations