Apple's firmware updates that were released yesterday have been installing properly for most people; however, a few people have run into problems when trying to apply the patch. After running AirPort Utility's firmware update option, the utility will fail and show a fairly generic-sounding error that states "An error occurred while updating the firmware."
If this happens, there are several things you can try to get the firmware to update:
Update AirPort Utility
Installation of the latest firmware update is recommended to be done through the latest AirPort Utility release (currently version 5.5.2). Check the current version you are running by going to the AirPort Utility menu within the program and choosing "About AirPort Utility." If the version shown is less than 5.5.2 then you should download the latest update. This can be done by choosing "Check for Updates..." in the AirPort Utility menu, or by downloading the software directly from Apple's Web site.
Before downloading, be sure to remove all currently installed versions of the utility. Though the utility should be kept in the /Applications/Utilities/ folder, some people may have multiple copies installed on their systems if they have manually moved them out of the Applications folder. Spotlight may be able to help locate and remove the utilities, which can be manually dragged to the trash and deleted.
Remove all previous firmware updates
AirPort Utility will store the downloaded firmware updates on your Mac in a folder within your user library. If for some reason the firmware update has been corrupted when downloading or is inaccessible to the utility, then you may get an error.
Go to the /username/Library/Application Support/Apple/AirPort/Firmware/ folder and remove any files within this folder. If you want to hang on to older firmware updates, you can just move them out of this folder, but the other option is to just delete them.
When the folder is empty, get information on the user library (/username/Librarary) by right-clicking it and choosing "Get Info" (or by selecting it and pressing Command-I) and then click the lock at the bottom of the resulting information window. Authenticate and then ensure your username (the one with "Me" next to it) has both read and write permissions for this folder. When this is set, select "Apply to enclosed items" from the gear menu at the bottom of the window. This step will ensure AirPort utility can properly access the library.
Remove Airport Utility preferences
In addition to ensuring you have the latest version of AirPort Utility running, remove the utility's preferences file, which is used to store some relatively benign settings for the program. This file is called "com.apple.airport.airportutility.plist" and is located in the /username/Library/Preferences/ folder. Removing this file will not harm anything, and its removal will result in AirPort Utility using default settings, which may help fix the problem.
The last option is to reset the AirPort device itself (including Time Capsules), which can be done in one of two ways: the first is a soft reset, and the more complete one is a "factory default" reset. The factory default resets will erase some settings on the device, so only try this if a soft reset does not work (steps are from Apple's knowledgebase):
Connect the base station to power.
Push and hold the reset button with a straightened paperclip for 10 seconds.
Release the button, and the base station will soft reset.
Unplug all connections and power from the base station.
Press and hold the reset button with a paperclip, then plug the base station back in while continuing to hold the button for approximately 5 seconds until you see the three status lights glow.
Release the button. The base station will factory default reset.
Again, first try a soft reset and then apply the firmware update, followed by a hard reset if it still is not working. You might also reboot your Mac and run AirPort Utility with no other applications running. There should not be any interference from other applications, but the possibility is always there.