I don't really think it's worth trying to rely on a tool, even if there is one. It's best to plan this out from the start, and code the layout optimally for all scenarios. The HTML on that page looks more complicated and awkward than it needs to be, which makes it harder to readjust for a different screen size. You really have two boxes there, which ideally should be at the same level in the HTML and perhaps floated. That way, it's easy to unfloat them at a certain screen width, so that one sits above the other.
Responsive design is a bit of a game changer for the web, because it forces us to think more carefully about our page structures. When you are only thinking about big desktop screens, it's easy to fall into the trap of pinning things to the screen with position: absolute etc., but that presents a lot of problems when you are trying to make the page more adaptive.