It looks like your having trouble getting the Slingbox connected to your new Airport Extreme. Your network settings for the Slingbox will need to be updated, so it knows how to connect to the new router. Please follow the steps in the link below to update the network settings in the Slingbox.
I hope this helps!
Sling Technical Support
4-5 months ago, I followed your instructions (as I had a new router), and my Slingbox 500 worked perfectly as well as on my iPhone; however in the last week or two, I can't get Slingbox to work on either my iMac or my iPhone. My iMac gives me a message that either my operating system or browser is not supported--or both. BUT, when I check them vs the Slingbox requirements, I DO have a proper OS--macOS Sierra 10.12.1 and a proper browser;Safari 10.0.1, On my iPhone, I get an error message saying to check my internet connection and that all devices are connected. My iPhone, iMac and my Slingbox are all connected to the internet and the same network. The iPhone message also said I could try powering off the Slinbox for 10 seconds then back on. I did that as well, BUT I still cannot connect to my Slingbox. Any ideas? Help! I really need Slingbox while I am on the road.