Is there a way to run a method/class only on Tomcat/Wildfly/Glassfish startup?

You could write a ServletContextListener which calls your method from the contextInitialized() method. You attach the listener to your webapp in web.xml, e.g.

<listener>
   <listener-class>my.Listener</listener-class>
</listener>

and

package my;

public class Listener implements javax.servlet.ServletContextListener {

   public void contextInitialized(ServletContext context) {
      MyOtherClass.callMe();
   }
}

Strictly speaking, this is only run once on webapp startup, rather than Tomcat startup, but that may amount to the same thing.

You can also use (starting Servlet v3) an annotated aproach (no need to add anything to web.xml):

   @WebListener
    public class InitializeListner implements ServletContextListener {

        @Override
        public final void contextInitialized(final ServletContextEvent sce) {

        }

        @Override
        public final void contextDestroyed(final ServletContextEvent sce) {

        }
    }

I'm sure there must be a better way to do it as part of the container's lifecycle (edit: Hank has the answer - I was wondering why he was suggesting a SessonListener before I answered), but you could create a Servlet which has no other purpose than to perform one-time actions when the server is started:

<servlet>
  <description>Does stuff on container startup</description>
  <display-name>StartupServlet</display-name>
  <servlet-name>StartupServlet</servlet-name>
  <servlet-class>com.foo.bar.servlets.StartupServlet</servlet-class>
  <load-on-startup>1</load-on-startup>
</servlet>