It seems to be a bug in Outlook 2013: It doesn’t show the “New email” or “New …” when you right click on the taskbar.
To solve this do the following:
- Unpin the Outlook 2013 icon from the Taskbar.
- Exit Outlook.
- Delete the following registry key: HKCU\Software\Microsoft\Office\15.0\Outlook\Setup
- Open Outlook again and re-pin the icon.