I just installed Fedora Core 4.

How do I find out what java version, if any, is installed?

In keeping with the Fedora Core 4 Release Notes, I have downloaded
java-1.5.0-sun-compat-1.5.0.06-1jpp.i586.rpm from jproject.org and
jdk-1_5_0_06-linux-i586-rpm.bin from java.com.

Now what do I do?

Question 1
man rpm

Question 2
google "how to install rpm fedora"

ok well firstly FC4 doesn't include java, so "man rpm" will tell you sweet fa. Personally i found installing java on fc4 a right pain in the arse. there are a lot of very detailed howto's online about how to build your own RPM from the original binary... but it's just far far too much hassle. instead i just got the .bin from java.com and ran it "sh jdk.whatever.bin" and it did the rest. it's not pretty or manageable comapared to an rpm, but it worked. I assume though that that rpm.bin file you found is new maybe.... just run it in the same way.