When migrating to Exchange 2013, all Mail is migrated to the Administrators Mailbox, rather than the users mailbox. You are running NME with Outlook 2010.
In order to resolve this issue you should patch your Outlook 2010 Client to Service Pack 2. The minimum system requirements for Exchange 2013 for an Outlook client are;
Refer to Microsoft Article - " Exchange 2013 System Requirements | Supported Clients "