Setup additional user called 'Share' and share out that users Contacts, Calendar, or Inbox. Having said that, if you are wanting to be able to setup a Contacts list and email to that list from Outlook (using To:), then you'll need to setup a Contacts list in SharePoint and connect to it from Outlook. The reason for this setup is that you cannot setup another user's Shared Contacts to be accessible from the "To:" command in Outlook. Wish that was possible but not yet.
Another reason I had to go this route was due to a messy export/import from a previous Public Contacts list to a SharePoint Contacts list - the migration does not match up column to column and can be a nightmare. Yet if you export/import from Public Contacts list to Exchange Online 'Share' user Contacts list, then you'll not have any issues.