If your Outlook for Mac inbox isn't syncing properly, you can usually resolve it by clearing your inbox cache.
1. Right-click (or control-click) your Exchange Inbox
2. Click Folder Properties
3. Click the General Tab
4. Click Empty Cache
Note: This will clear out any locally cached emails and re-sync your email application with the Exchange server. This process will take several minutes and appears as though nothing is happening. Rest assured, your local cache is being cleared and when the process is complete you will be able to click the OK button.