bcrypt is not available for PHP 5.3.x. You need to use the library from ex. https://github.com/ircmaxell/password_compat
You don't need to install anything on the server and you don't need to have root access. Just install the package and start using the library :)
You can either use https://getcomposer.org (Dependency Manager for PHP) or just download and include it in your project.
When you upgrade your server to php 5.5 you can use crypt out of the box, as it has been implemented as part of the language.