How to restore Airport settings?
The stored passwords for Wi-Fi networks (as well as many other things) are stored in your keychain file.  the easiest way to get stuff out of this is if you have a MobileMe account which has the option to synchronize the keychain, so it will appear automatically just by logging in to MobileMe.

There are 2 drawbacks to this:


You need network connectivity to login, and so if you don't have an available network because you don't have the key you are out of luck,
MobileMe is deprecated, and the functionality is not subsequently available through it's replacement iCloud.


You could always boot from your Time Machine backup, open up the Keychain Access command and export the keys you want.  They will be stored as files within the backup, but are encrypted so not a lot of use. It may be possible to import the keychain if you can find it (I'm not at a Mac right now so can't check for location) and you know the administrator password for the system is came from, but I'll leave that for another user to comment on or provide an answer for.