Under settings > email+accounts > microsoft account there is no contact box only Email.
Email box is checked by the way. So I tried to add a new contact from the website outlook.com and the contact is visible on the phone.
So the sync between the phone and microsoft account works but I can not able to save contact from phone to outlook.
On my college's phone (same os version but another vendor) the save to option is visible when he tries to add a new contact