As much as anything I think this is actually an analysis and requirements issue.
You say your client wants you to run Tomcat on port 80. Is this
really what the client wants, or do they just want a URL for your web application which doesn't have a port number in it ?
If they just want a simple URL, then
you should be able to put a little port-redirector on another machine (or in your firewall/router etc) which redirects all traffic for port 80 to port 8080 "behind the scenes".
Alternatively, if you already have an Apache running on port 80, you should be able to configure it to hand off all or some of its traffic to Tomcat (see the Tomcat web site for details of integration with Apache).
As another alternative,
Java is portable, so do you really have to run your web application on a Linux machine? If you have admin skills for another operating system with Java support (Windows, for example) why not put Tomcat, and thus your application on port 80 of some spare Windows machine.
To offer more detailed advice we really need to know a bit more about what the real possibilities and constraints you are facing boil down to.