Almost everything in the book, and most of the code examples, applies to Java SE 6 as well as Java EE 5. That is because Java SE 6 includes JAX-WS and JAXB.
Besides JAX-WS and JAXB, the code examples from the book use JSR-181 (Web Services Metadata) for annotations. The only Java EE 5 specific technology used in the book is related to JSR-109 which specifies Java EE 5 deployment standards.
So, any platform that supports JAX-WS, JAXB, and JSR-181 can be used to implement the code examples and techniques described in the book. This includes platforms (with the appropriate plug-ins to provide JAX-WS, JAXB, and JSR-181): * Tomcat * Spring * XFire * Apache CXF * etc.
The only caveat here is that each platform is slightly different with respect to deployment, so you might need to tweak the example to deploy in each of these.
On my blog, there has been discussion about porting the example code to Tomcat. Please post a comment there if you have ideas/suggestions.
Having said all that, I would strongly disagree that Java EE is "dying". The major app server vendors are bringing out support for Java EE 5. BEA is already there with WebLogic Server 10. JBoss AS 5.0 will be Java EE 5 certified and is available in beta now. And, of course, GlassFish has been hugely successful as the first Java EE 5 certified application server. It has been out for over a year.
The real problem with Java EE is that it has been so successful that it is getting to be too big for a lot of people. Java EE 6 will address that by allowing vendors to provide distributions for specific markets. This will allow vendors, for example, to provide a "Web Profile" version of Java EE 6 - just targeting Web development - and including only the features needed for that market.