Can you really write an easy python script which handles real-time server pushing of information to the customer when information is available?

"Yes, but":

You can preserve the bond between your client and server live but this could take considerably more server assets.

Edit: If you wish to do that, see

You could have the customer register an Ip that the server can contact, however, you will encounter a lot of challenge with client-side fire walls.

Or simply do what most services do, and also have the client poll the server from time to time.