To save Outlook contacts to your native address book:
First go to Device Settings > Contacts and confirm that the 'Default Account' is set to 'iCloud'. If you do not see this setting that is OK, the setting should only be available if you have more than one account added to your device. If there is no account connected to the device, then your Outlook contacts will be saved locally but will not be synced to other devices.
Second, open Outlook for iOS and in the left nav view open the Settings gear > Select your account > Toggle ON the 'Save Contacts' toggle.
Saved contacts will be added under the contacts in your iCloud account. These contacts will sync to all devices which are signed in with the same iCloud account.
Note: You can only export contacts from one device at a time.
If a user has contact-export turned on both their iPad and iPhone one will need to be disabled so that we are able to manage conflicts. Duplicates can take up 24 hours to be reconciliated.