No, the new law has no judicial review and has a few other things that wouldn't fly in the US. It's markedly worse (though don't get me wrong, the US definitely has it pretty bad in this area too).
You say that like FISA Courts are actually judicial review and not rubbing stamps... where you win is that you have a stronger set of rights and case law about it.
The difference is that there isn't even fake judicial review. And I disagree that we have a stronger set of rights -- the difference is that the NSA explicitly ignores your constitutional rights.
All of our rights (other than the right to a jury for certain criminal trials, freedom of religion, the aquisition of property must be 'on just terms', the right to be a senator if you can vote, and the right to vote in federal elections) are in common law. This means that any new law can overturn those interpretations.
Personally I think Australia needs to push for a constitutional bill of rights. Unfortunately this is going to be a very hard battle to win, given the enormous requirements to get a constitutional amendment passed.