The router configuration is only to enable public access to the Slingbox. This is necessary for remote viewing. You should be able to find instructions from this website (http://portforward.com/english/applications/port_forwarding/SlingBox/SlingBoxindex.htm).
As for your Slingbox, you should perform a factory reset (http://support.slingmedia.com/get/KB-1000038.html) to clear the network configuration from the old router. This will allow it to get a brand new IP address from the AirportXtreme.