Class to simplify skipping some actions on the first loop iteration
Don't try to over-generalize early. This is a situation I've run into quite a few times for strings, but never anywhere else, so I would recommend you start with a string-specific solution, and only try to come up with something more general if it ever becomes necessary. 

You mention String.join- is there any situation where you face this problem where String.join wouldn't solve your problem? It's quite possible that the problem you actually need to solve isn't "skipping some actions on the first loop iteration", it's "cleanly joining strings with a separator". If that's the case, then why not just reimplement String.join yourself, rather than creating your own very different way of achieving the same thing? 

If you do need to generalize, then your solution could work, but its main problem is that now you're creating a class whose only purpose is to run a chunk of some arbitrary (possible private!) procedure, just because that chunk happened to follow a particular structure. If Java 6 supported functional programming, then a single class which was passed functions would have been acceptable, but the need to create a new class that inherits from FirstCartridgeIsBlank every time you want to use it seems like a design headache to me.

So I think instead of trying to create a class for this, it'd probably just be simpler to do it by extracting methods:

public static String prepareUrl(String url, List&lt;? extends NameValuePair&gt; arguments) {
    boolean firstIteration = true;
    StringBuilder urlBuilder = new StringBuilder(url);
    for (NameValuePair argument : arguments) {
        String argumentString = getQueryArgument(argument.getName(), argument.getValue(), firstIteration);
        firstIteration = false;
    }

    return urlBuilder.toString();
}

private static String getQueryArgument(String name, String value, Boolean includeSeparator) {
    //...
}