I just switched to CakePHP1.3 to CakePHP2.2.3.
Regarding controllers, the old app/app_controller.php was moved to app/Controller/AppController.php.
It's the same for models, app/app_model.php was moved to app/Model/AppModel.php.
The Controller and Model folders start with an uppercase letter now.
Also, the controller/model names are not using _ anymore and are named using uppercase letters: instead of app/controller/users_controller.php, you will have app/Controller/UsersController.php.
My best advice would be to start with an empty CakePHP2.2.3 project and copy your older project files in their corresponding folders.
You can also take a look at the migration guide here:
http://book.cakephp.org/2.0/en/appendices/2-0-migration-guide.html
There are some improvements in CakePHP2 that you should know about (like the lazy loading of classes which improves performance, deprecated components, etc.).