If you arent in a rush and have Outlook.com, you can let the server make a backup - the setting is in Settings, under General. But it can take up to 4 days. It doesnt delete the mail though, so you'd still need to empty the folders, one by one.
If you are in a rush or want to have some control, you can use Outlook's Autoarchive. This will move and delete. You need to set it to archive all mail older than 1 day. This will move it all to a pst file and delete it from the server. You may have some mail you'll need to manually move, but this will handle the bulk. If you only needed to do one or two folders, Export then manually delete can be almost as fast.
Using Outlook's AutoArchive Feature
After running autoarchive, check Calendar, Contacts & Tasks - Contacts need to be exported or moved manually and some appointments and tasks may need to be moved or exported.
After its finished, check the deleted items, deleted recovery, and junk folders in web access.
As an FYI, Autoarchive may be disabled on some business Exchange accounts - this is automatic if you have an online archive.