You can tap on either an Airport marker in a displayed route or on a map data marker (eg: Flight Rules, Temperature, etc). Doing so will bring up the iPhone-style Airport page. Tapping on 'Details' will take you to the listing on the Airports page.
You can also tap and hold on an airport shown on the map. From the pop-up, select the 'Airports' button on the bottom and then next to the airport you want to view information for, tap on 'More' and then 'Details.'