First of all, this AirPort feature relies on a working "dynamic DNS" setup. If you let someone else host your domain and manage your DNS, get them to set it up and get the TSIG key name and private key for updates. If you manage your own, first follow a tutorial on configuring dynamic updates in your DNS server.
Afterwards:
In the (dynamic global) Hostname field, enter the fully-qualified hostname (e.g.
whitebrick.example.com). Note that despite what you'd expect. even if you've set a domain in the TCP/IP tab, it does not apply here, so enter a fully-qualified name please.This is the tricky part!
To avoid the AirPort Utility limitation (with the '.'), type the hostname you desire for your AirPort elsewhere (e.g. in Notepad if you're on Windows, or TextEdit on OS X), then copy and paste it into the (dynamic global) Hostname field. (This has not been tested on OS X, but only on Windows.)
In the User field, enter your TSIG key name. If you use a managed DNS service, you should have got it from your service provider. If you're like me and use ISC BIND, this is found in the
K<...>.keyfile:alice.example.com. IN KEY 0 3 157 ...In this example, the username is "alice.example.com" (without the terminating period).
In the Password field, enter your TSIG private key. With ISC BIND, this is found in the
K<...>.privatefile:Key: MQ0KMDA6MDE6NTEsMDkxIC0tPiAwMDowMTo1NywzOTQNClRoZSBEb...
In this example, the long key is the password.
Restart the AirPort and it should register its hostname.