Hi, welcome to Cologne
And welcome to the jungle of German bureaucracy
Usually, if you have a car that's authorized in another country, and if you want to
drive through a "Umweltzone" (environmental zone) in a German city, yes, you need a badge.
For Lanxess Arena there are several ways where it's allowed to drive through without a badge.
I hope you can somehow understand this website, for me the English translation somehow
didn't work correctly:
The key is at the bottom of the passage: "Die Zufahrt zur LANXESS