If using an iPhone and Outlook on the same email account, you should make sure that you set Outlook to "Leave messages on server", ideally for 5 days.
In general, any email sent from one of your devices, will not appear on any of your other devices (unless you've cc'd yourself). The iPhone can be set to do this automatically.
If you have LMOS (Leave Messages On Server) setup correctly, you should receive a copy on both devices - not quite the same thing as being in sync, but we believe this should help satisfy your query regarding incoming emails.
For further support on this, we would suggest that you contact Apple/Microsoft directly or search their FAQ/Help sections.