Unfortunately, emails sent to your
appspotmail.com are not saved anywhere, so you can't really access them through a client.
One thing you could do is to make your handler get the email, address it properly (starting up your instances, etc), then create a new email with the same content and send it to an address where you CAN read it (like your personal gmail, or an admin email). I would suggest using the
Mail Python API (if you're using Python), which you're possibly already familiar with since you receive your emails there.