Usually, you don't have to check for these particular libraries. You certainly don't have to upload them. What you check is if the same version of the JRE is installed in your server. You are using 1.5.0_22. If it is installed in the test server, it's enough.
The JRE libraries are part of the JRE itself, so the project does not need to (and SHOULD NOT) include them explicitly in the deployable WAR. Eclipse is using its own local JRE and the libraries in that JRE to do local compiles and other work, but once the app has been deployed, it's the responsibility of the server to provide them.
Unless you are taking the unusual step of augmenting your JRE with custom libraries - which is a bit of a bother, and semi-dangerous as well - this is not something you need to worry about. The default situation is that everything was already done for you.
An IDE is no substitute for an Intelligent Developer.