If you specifically want the URL mappings, there are a few ways, but they all require some information from the deployment.
For example, if you know the name of the Servlet, you can use ServletContext#getServletRegistration(String)
ServletContext context = ...;
Collection<String> mappings = context.getServletRegistration("servlet-name").getMappings();
If you don't know the name, you can get them all with ServletContext#getServletRegistrations()
Map<String, ? extends ServletRegistration> registrations = context.getServletRegistrations();
and try to find yours, maybe by comparing classes (your servlet class versus the class name from the ServletRegistration).
Note that you will still probably have to try matching your current request's URL to the Servlet's url mappings to be sure. You'd have to go to the Specification to find out how the mappings actually work.