Organizing and accessing contact information in Outlook.com
Here's a work around. After you update or add a new contact on your iphone, go back to the main Contacts screen. It's the one where you can search for a contact and it says Groups in the upper left corner of the screen. Tap on Groups. On the Groups screen swipe down on the screen. You should see the circular type hour glass go around a few times. Your phone just updated iCloud Contacts with the change or add you just made.
You shouldn't have to do this work around as anytime you make a change or add, your phone should automatically update iCloud Contacts. Calendar still seems to be able to do the update automatically. But unfortunately for some reason Contact updates stopped many months ago and Apple doesn't see to get it.