How can I copy a message folder from one mailbox account to another in a different domain using OWA?

The Outlook Web Access client does not provide a way to backup email folders; you will need to use Microsoft Outlook or Microsoft Outlook Express to create a PST / Data file that will be saved to your computer.   

If you are working from a remote location your Exchange Administrator will need to provide you access to your mailbox via a VPN connection or RPC over HTTP.  Once you set up your account using Microsoft Outlook, you can refer to the link below for more information on how to create an Outlook Personal Folder (.pst).

How to create a .pst file in Outlook 2003

 Note: If you are using Microsoft Outlook 2007 you will use the File\Data File Management option. 

Once the PST is created, you can then set up and Outlook connection for your other email account and open the PST file.  You can then simply drag the mail items out of the PST and into your mailbox.