You should be able to do this very easily by following the steps they have provided.
This series of answers should lead you in the right direction on setting things up with your domain registrar/host
There are plenty of tutorials on other blogs that could also help you with this.
Login in to Google Apps for your Domain through http://www.google.com/a/cpanel/yourdomain.com
Go to "Domain Settings" > Domain Names
- Click "Add a Domain Alias" and follow instructions.
If you want to allow "Google Sites" to publish to your domain or a subdomain, go to the Dashboard and select "Sites" and you should be able to change the URL at which they are accessed once your domain is verified (or possibly before, I'm not sure).
As to the question of whether it is possible to actually use Google Sites with your own domain with a free Google Apps account, I can assure you that it is and my own Google Sites can be found at http://sites.bjtitus.net (although they are all private).