Alternately, you can install a prebuilt .deb
There is air 2.6 (last supported Linux one) available here for both architectures (Both use 32 bit underneath but at least it works). Thanks to prusswan for pointing it out.
Make sure you have 32 bit libs it'll need:
sudo apt-get install ia32-libs
I downloaded the AMD64 one (adobeair_18.104.22.16870_amd64.deb) and installed on my computer (am64 with Precise Pangolin 12.04)
sudo dpkg -i adobeair_22.214.171.12470_amd64.deb
Then I installed the application that needed Adobe Air in the first place and it worked.
Update for Ubuntu 13:10+,
Trying this technique, i get "Package ia32-libs-gtk is not installed"
I looked up the package and tried installing those dependencies and other related packages I can think of directly:
sudo apt-get install gtk2-engines:i386 libart-2.0-2:i386 libcairo2:i386 libcanberra-gtk0:i386 libdatrie1:i386 libgail-common:i386 libgconf2-4:i386 libgtk2.0-0:i386 liblua5.1-0:i386 libpango1.0-0:i386 libpixman-1-0:i386 libqt4-network:i386 libqt4-test:i386 libqtcore4:i386 libthai0:i386 libbonobo2-0:i386 libglade2-0:i386 libgnomecanvas2-0:i386 libidl0:i386 liborbit2:i386 libwmf0.2-7:i386 gtk2-engines-murrine:i386
But that didn't work for either the deb or original binary.