Great! Thanks for the update.
Out of morbid curiosity I went to my Mountain Lion Mac and downloaded AirPort Utility 5.5.3. I confirmed it will not install on Mountain Lion using OS X's usual package installer, but using the technique that follows I was able to install and run it on Mountain Lion. The only thing I could not verify is its ability to administer an older AEBS since I no longer have any to test.
The remainder of this post is for those who are willing to circumvent Apple's desire to use only current versions of AirPort Utility for reasons known only to them. Proceed only if you accept this as an unsupported workaround strictly for the purposes of administering an older Extreme on a newer Mac.
Prerequisites: A package extractor like Pacifist or unpkg will be required. Pacifist is shareware, unpkg is free.
Download Pacifist here: http://www.charlessoft.com/
Download unpkg here: http://www.timdoug.com/unpkg/
Download AirPort Utility 5.5.3 here: http://support.apple.com/kb/dl1390
Directions for Pacifist (I do not have screenshots for unpkg but the idea is similar):
After the .dmg file finishes downloading double-click it to mount the volume containing the metapackage. Control-click it and select Open With > Pacifist from the contextual menu:
You will get the following dialog:
The "unidentified developer" in this case is Apple. Click Open to open Pacifist.
Click the "reveal triangles" until you see AirPort Utility.app and select it.
The next steps are important.
Select Extract To... from the toolbar as in the above screenshot. In the dialog that follows, choose Desktop since otherwise you will overwrite your existing, newer version of AirPort Utility in your Utilities folder which should not be deleted. Restoring it may not be trivial.
Authenticate, allow Pacifist to extract the app, then you can quit Pacifist.
You will now have AirPort Utility version 5.5.3 on your Desktop. Before you attempt to copy it to your Utilities folder though, be sure to rename it accordingly:
You can then drag it to your Utilities folder. Authenticate, and you will have it alongside later versions where they can peacefully coexist:
Use the newest version AirPort Utility to administer current production AirPort Base Stations, and older versions only as necessary for older AirPort Base Stations that cannot be administered any other way.