MICU (Microsoft Installer Cleanup Utility) might work, but it's something I've never used.
Regarding System Restore. I always tell people that System Restore is an absolute last resort if Windows isn't working properly. In other words, if your only other option is to reformat and reisntall the Operating System, try Restore first. Other than that it does more harm than good 99.99% of the time.
If you're comfortable with REGEDIT, you can manually delete the Reader folder from Program Files, and then delete all keys for Reader 9 from the HKEY/LOCAL MACHINE/Software/Adobe and HKEY/CURRENT USER/Software/Adobe registries, reboot, and then try re-downloading and reinstalling Reader. Again, that's IF you're comfortable with editing your registry. If not, go with the MICU.
I'm not familiar with that process.
Now I installed the windows installer cleanup utility. I was able to remove Adobe 9.3.1 but now the download manager. I restarted my computer and tried to installe acrobat reader again. The same message appears: The following application could not be installed: Adobe Reader. What is the problem?
A few things I'd suggest:
Clear cache, history and cookies from your browser.
Try redownloading the Reader again and install from the new download instead.
Last resort: try installing in Safe Mode.