No, "zaraz" is more like "imminently" or "after a moment". "Instantly" would be closer to "teraz".
It sounds a bit like that in Russian as well, but it also sounds a lot like "infection"...
"Zarazek" in polish is "germ", and it's a diminutive form of "zaraza" - "pestilence". Some people joke that "zaraz" is a big bacterium. Also it's not "instantly" (natychmiast). It's a little later than instantly, like "as soon as I finish doing what I'm doing".