If you do not have access to a mailbox within Outlook (to facilitate a simple export to PST) you can do so via PowerShell.
Firstly create a shared folder (must be accessible via UNC and not simply a local or mapped drive) and assign the “Exchange Trusted Subsystem” a minimum of RW on it.
Within PowerShell (EMS) to gain access to the necessary tools you first have to assign the correct rights to the user account (presumably your admin one) that you are using to connect to Exchange:
New-ManagementRoleAssignment -Role “Mailbox Import Export” -User <adminuser>
Close and restart the Exchange Management Shell.
Run the command to begin the export of the mailbox:
New-MailboxExportRequest -Mailbox <username> -FilePath \\server\share\username.pst
The export may take a while but you can check on progress by monitoring the file in the share or by the PS command Get-MailboxExportRequest