Organizing and accessing contact information in Outlook.com
Even Apple couldn't fix this, basically you will get an email from Outlook in your inbox and it states
"ACTION REQUIRED TO SYNC CONTACTS"
Read this email and either use the Outlook app or say no to use the default mail app with your iPhone.
If you don't want the outlook app then hit no and then all your contacts will sync
> Passwords & Accounts. Tap Exchange, sign in to your Outlook.com account, then turn on Contacts.