Thank you to @Christ and @lembas.
First, I really think that were are just missing something here. I have
installed Ubuntu 12.04 before on this Macbook Air using the alternate image
and installing free software only (toogle options dialogue F6 at
installation). The specific image is called
ubuntu-12.04.2-alternate-amd64+mac.iso.
So, this has to work somehow*.
Using different hardware is of course the easier option. Since the two USB
Wireless LAN adapters that I have do not work, I will have to buy a new one.
However, the ones at thinkpenguin are very expensive
(https://www.thinkpenguin.com/gnu-linux/penguin-wireless-n-usb-adapter-gnu-linux-tpe-n150usb).
I'd rather buy it somewhere else and donate 20$ straight to the project
instead.