There are several ways to achieve this, but the most common way to solve it is to run Apache as a reverse proxy in front of it. You can find some details here. This will work on both Linux and Windows. For Linux, you can also use authbind to allow Tomcat to bind to port 80.
Just changing the port to 80 in your server.xml will not work in Linux, since it would require you to start Tomcat as root, which is not a very good idea.
Also, to have your webapp at /, you can deploy your war file as ROOT.war.
