Interesting...as I am running 10.1.1 I tried the Software Update from my 9.2.1 Classic and it said that the software didn't need updating.
So, I downloaded and ran the updater (which is a Classic app) under Classic on 10.1.1. It seems to have run OK, though it means business when it 'quits all open applications' as all my OS X apps, including OS X Finder, quit while it ran. I had thought I would be able to keep using OS X (maybe play a game) while the Classic update was going on! I had it update my Classic system folder, but then it wanted to Restart my entire Mac! No need for that! I used Force Quit on the installer, got back my OS X finder, and found that Classic was no longer running. Fine, since I needed to restart it anyway.