No, Tomcat isn't an EJB container. Only Glassfish 3, JBoss AS 6, etc are. For Tomcat you have to install it separately. The linked answer was just a basic kickoff example. You can do it as good without EJB. You only need to create it yourself during bean's initialization, construction or postconstruct. E.g.
private UserService userService = new UserService();
or
private UserService userService = ServiceFactory.getUserService();
etc.
Note that this isn't existing code. It's just code which you have to write yourself. All it contains are just methods which does all the database interaction task. In an EJB capable container you'd use JPA for this. But on a simple servletcontainer like Tomcat you'd need good ol' JDBC for this. You can find extensive kickoff examples of a basic JDBC DAO in this blog.