Public folders won't sync to smartphones, but what about site mailboxes? Can they sync?
Site or shared mailboxes don't have their own logon but they can be opened (as the only account) in an Outlook profile using the credentials of anyone who has Full Access permission. Unfortunately, when you use your credentials to set up an Exchange account on a smartphone, it opens your mailbox, not the shared mailbox.
The shared mailbox cannot be hidden from the address list when you are setting up the account. It needs to be visible for this method to work. After the account is configured, the address can be hidden.
However, you can open the shared mailbox using POP3 or IMAP (if those services are enabled for the shared mailbox) using youremail@address\sharedmailboxalias format for the login. Some servers may require domain\useralias\sharedmailboxalias format.
For example, this configuration works on an iPad/iPhone (and also on Android and Windows Phone) to connect to an Office 365 shared mailbox.
When you configure the account as POP3, messages are downloaded and marked as read on the server. Use IMAP if you want to keep (and sync) the read/unread state.