I was looking forward to try out the new site mailbox app in SPO 2013. I was assuming it could solve my problem of being able to send an email to the mailbox with a document attached to the email after which I could route the document to a SharePoint library for further documentprocessing. However, now that I have this feature and trying it out I cannot figure out how to configure that scenario. Is my scenario possible? How do you configure that?
No, That is not possible.
The Site Mailbox in SharePoint is really just a Shared Mailbox in Exchange. Anyone that has access to the site can click "Site Mailbox" and have access to OWA for that Shared Mailbox.
There is no way for the document to get out of exchange and into SharePoint without you doing it.
You should look into OneDrive if you want to sync files to a doc library
The new Delve introduces a number of options for integrating mailboxes and SPO documents. If your tenant is set to "FirstRelease", Delve should be enabled there already.
For more information on Delve: