The article is saying that you don't "build for the older browser" --- or indeed any browser specifically, but instead start off with basic HTML and adding just enough markup and styling around the content as you desire. Maybe you'll later tweak the site to look exactly the way you want in a specific browser, but this process means that even if it doesn't look perfect, other browsers will likely be able to use the site too.

No. You have to build on the new browser, because that's what is on your new computer, then hope it runs on the old computer with the old browser.