providedCompile without war plugin

There is no such configuration out of the box for the java plugin. However you can build it yourself as follows:

configurations { providedCompile }

dependencies {
    providedCompile "javax.servlet:javax.servlet-api:3.+"
}

sourceSets.main.compileClasspath += configurations.providedCompile
sourceSets.test.compileClasspath += configurations.providedCompile
sourceSets.test.runtimeClasspath += configurations.providedCompile

This adds the configuration, and puts all the dependencies in there on the compile classpaths of both your main classes and test classes. You also need to add it to the runtimeClasspath, as that doesn’t include the compile classpath according to the gradle DSL documentation.