How to set Automatic Replies (Out of Office) on Shared Mailboxes using Office 365

By using the Outlook Web App, you can do this extremely easily. The steps are as follows:

1. Log into the Outlook Web App at https://outlook.office365.com using your own Office 365 credentials
2. Click on your profile image (or placeholder image) in the top right
3. Choose Open another mailbox 
4. Type in the name or email address of the Shared Mailbox and select it
    
5. Click the settings cog in the top right, then click View All Settings, then Automatic replies
6. Configure the settings and write your message, then you are done.