If your Mailbox Level backup is failing with this error ( Failed to sync changed items for folder "Exchange\Mailboxes\"domain"\Public Folder\"Public Folder Name" ) you will want to check the permissions for the "user" backing up these folders.
- Open Exchange Management Console.
- Click Toolbox.
- Double click on Public Folder Management Console.
- Click Default Public Folders to populate folders inside the console.
- Highlight the folder in question then click properties on the right.
- Make sure the user gets selected and it has these permissions Create items, Read items, and Edit all.
- Once complete repeat on all folders necessary and re-run Mailbox backup.