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 arena ist aber auch für alle Fahrzeuge ohne Plakette gesichert.
Alle Infos zu den Transitlinien entnehmen Sie bitte der Übersichtskarte der Umweltzone Köln (Stand: 1. April 2012) "
It says that there are ways to Lanxess Arena without any badge.
When you click on "Übersichtskarte der Umweltzone Köln (Stand: 1. April 2012)" you will find a map with the ways to the Arena
where you definitely don't need a badge. This map is a bit impractical, so I guess when you come from the Liège route, for you the
route down south around the city should be right. You take the A4, direction cologne south, cross the river, and then your GPS should guide you the right way
over A559 which ends directly in front of the Arena area where you can easy find several parking lots...
If you desperatly want a badge just because you want a nice German car acessoire, you can purchase it online here it's only 15€ xD https://www.tuev-sued.de/auto_fahrzeuge/feinstaub-plakette/bestellen/england
i'm not quite sure, but i guess it never expires only if you get a new car...
I hope it helps a bit, if you have further questions, feel free to ask