In Outlook Mobile settings it does not allow the enabling of "Sync contacts". Next to the option it displays the message "Contacts sync isn't turned on in system settings".

To enable it swipe down your Quick Settings menu and ensure the setting "Sync" is enabled. Or go to Settings -> Cloud and accounts -> Accounts and enable "Auto sync personal data" option at the bottom of the menu.

Then go back to Outlook Mobile settings and then it should allow you to enable the Sync Contacts option.