To access their archive mailbox using Outlook, users needs an licensed Outlook 2013 or higher. Sometimes an Outlook 2010 is also possible, detailed information can be found in this Microsoft Article.
How do I enable In-Place Archive for one user?
Go to the Exchange Admin Center
Recipients -> Mailbox
Select a mailbox and then you can enable Archiving:
Then the mailbox type is change to type 'User (Archive)'
It's possible you see this error below for a couple of hours. You can continue.
After a while (minutes or hours) the 'Online Archive' will be automatic pushed to the Outlook client as additional mailbox:
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