Will you need to re-compile the code? No. That's the beauty of Java's portability.
Does tomcat have a 64-bit version? The have a .exe that allows you to run Tomcat as a 64-bit service, which, if you tell it to use your 64-bit JRE and JDK should suffice. You download it from here: Tomcat 6 SVN. Since it isn't part of the normal Tomcat distributable I am not sure if it is supported and tested or not. I don't use it so I have no experience with it.
Another reason for not recompiling the code is that the environment information is not available in the code. All the memory management and execution is done by the underlying VM and the container. So as Steve says, try out the Tomcat 64 bit version.