For some reason, Outlook 2013 doesn’t seem to like Outlook.com accounts. Sometimes, for no obvious reason, you’ll encounter sync issues like emails that are not shown anywhere but are still already marked as synced by Outlook 2013. If that seems familiar to you, I may have a workaround.

Procedure:

1. Try to open you account on another application which do shows the emails.
   
2. Select all recent mails, or the mails that didn’t show up in Outlook 2013.
   
3. Move them to another, new folder.
   
4. Sync that folder in Outlook 2013. Your mails should appear there.
   
5. From Outlook, move all your mails there to your previous folder.

That’s it. I solved the issue for me without having to delete my Outlook 2013 OST and re-sync every email.