It might be much easier to simply "Export" all your configuration settings to a saved file on your Mac...and then "Import" them again when you need them.
Open AirPort Utility and select your AirPort
Click on the File menu in the upper left corner of the screen near to the Apple icon
Choose Export Configuration File, name the file and save it where you want
When you have a new router, open AirPort Utility, click the File menu and choose Import Configuration File, navigate to the location where you saved the file and click Import.
While it is possible to take screen shots, that will take a good deal more time, and you will still need to change all the settings manually when you want to configure a new router.