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. This will work with on-prem Exchange provided POP3 or IMAP support is enabled on the shared mail.
Server name: outlook.office365.com
Encryption method: SSL
Server name: smtp.office365.com
Encryption method: TLS
Enter the Shared mailbox display name and email address and something in the password field on the New Account screen and click Next.
Enter the incoming and outgoing server names, the username in youremail@address\sharedmailboxalias format and your password then click Next. The mailbox is now ready to use.
Screenshots from an Android tablet, showing the server settings for Office365 shared mailboxes are below.
Enter the shared mailbox's display name and SMTP address on the first screen. Type something in the password field. Click Next.
Enter the server name and port settings, the username in youremail@address\sharedmailboxalias format and your password. Click Next.
Enter the outgoing server name or port settings. If needed, use your username and password for Outgoing authentication.
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.