Organizing and accessing contact information in Outlook.com
I had the same issue. I was wondering the contact didnt sync. Then I saw there was only one message in my outlook mailbox on the iphone. This message also didnt appear on outlook.com.
In this message Microsoft tells you that they recommend Outlook for ios. They give you the choice to download the outlook app or start the sync with Apple mail. And if you look closly, you can see:
if you don't choose something in this message, the sync automaticly starts in some hours.
not the best way to recommend something..