Generally speaking, you should be able to complete update installation by shutting down. Usually, when doing an update, it gives you the option to immediately restart or keep work and restart later. If you choose the second option and then shut down, under normal circumstances, it should finish installing the updates in my experience. If you cannot, then it might be problem with the update.
The other overall way to install updates is to manually download and install them. You can find those updates on Apple's download page. Here is the page for Mac OS X updates:http://support.apple.com/downloads/#macos
As to why the problem is happening, it could be a corrupted update (i.e. corrupted somehow during the download), your Mac OS installation is corrupted, or maybe just some preference is corrupted.
For the first, you can try downloading the updates that the Mac OS says you need from the link above and see if they will download and install.
For #2 or #3, the first thing to try would be create another admin user account and see if the things work better there...or first logout of the current user account and then try restarting. Next you could try doing Verify/Repair Disk Permissions or the Disk itself using Disk Utility...it likely won't do much, but it is worth a try.
The ultimate option would be to re-install the Mac OS...but let's try other things before we get to that.
BTW, what version of the Mac OS are you using?