Different to what some of us had hoped, the SDK update process didn't get much easier with Flash Builder 4.7. 😐
Furthermore, a fresh install contains the already outdated AIR 3.4, which produces lots of compiler errors (at least on my system). Thus, you should upgrade to AIR 3.5 directly after the installation.
And this is where Adobe created two stumbling blocks for us:
1) You have to download the right SDK! Yes, now there are two, and you have to pick the one that includes the new ASC 2 compiler.
2) The SDK needs to be put to a different place now.
So, after installing FB 4.7, get the latest AIR version from here:
Or from the new Gaming SDK. But NOT from the usual place.
Extract this archive, and copy it to the following path (OS X):
/Applications/Adobe Flash Builder 4.7/eclipse/plugins/com.adobe.flash.compiler_220.127.116.119722
(I guess in Windows it's "c:\program files\Adobe Flash Builder 4.7..." instead. Please correct me if I'm wrong!)
Now, this directory already contains a folder "AIRSDK". This is the old version. Rename it to e.g. "AIRSDK_3.4". Then rename the new SDK to "AIRSDK" instead. Make sure that the contents of those two folders look the same; if they are, you have set it up correctly.
Now start up FB 4.7, and it should use AIR 3.5/3.6 (whatever you installed). You'll also need the latest Flash Player, to be found here (3.5) or here (3.6 beta).
That's it! Good luck, everyone!
Adobe has created an official documentation on how to upgrade the SDK. Find it here: http://helpx.adobe.com/flash-builder/kb/overlay-air-sdk-flash-builder.html