Office 365 supports the ability to create Shared Mailboxes, right through the UI or through PowerShell commands. The Shared Mailbox is essentially an unlicensed mailbox with no direct logon capabilities. This means you cannot open the mailbox directly, within the Outlook client. What you can do is open the mailbox with an existing licensed mailbox, as long as you have permissions to the Shared Mailbox and at the right levels.
What this leaves you with is the inability to create specific Outlook Rules. You can create standard OWA rules through the Office 365 admin portal or by opening the mailbox from within OWA (if you have full permissions to that mailbox.) Most of the OWA rules are basic, but you can achieve typical Shared Mailbox processes that most organizations put in place. (e.g., you can move items to various folders, as they arrive.You can also forward messages to certain individuals within the organization, etc.) What you cannot do is setup an auto-reply rule. As an example, you may have a “sales” alias for customers requesting certain information. You may want that mailbox to send an auto-reply, stating that you received the message and someone will get back to you in X hours.
Since this is a Shared Mailbox and you cannot log on to it directly, you are unable to create that rule. So how do we achieve this, if the rule doesn’t exist? The only way I have seen this work is to create the rule directly in Outlook. The only way to logon in to the Shared Mailbox with Outlook is to license the mailbox directly. Chances are you have many Shared Mailboxes and they are all mailboxes that will not need an auto-reply rule. So the path I would take is to license the Shared Mailbox in Office 365, for only the mailbox that requires the auto-reply rule. I see the next step for Microsoft is to add this option to OWA or give away a free license for these specific scenarios.