Q: Get error message about downloading firmware update for AirPort

What does the error message say, exactly? Which version of OS X are you running on that machine?

For the firmware updates to work, you need to be connected to your AirPort device and connected to the Internet at the same time. If you have some situation where you have to disconnect from the Internet in order to connect to your AirPort device, it's not going to work, at least not for the first time you need to download a given firmware version for a given model of AirPort base station.

Apple's servers are overloaded.  Wait a few hours, or try again tomorrow.

I doubt that's it. Apple seems to be serving these updates via Amazon's CloudFront CDN. It's more likely that OP's local CloudFront nodes don't have the updates yet, or something.

My update downloaded, then started to update, then stopped and gave me an error message that the firmware update failed.  Now my Time Capsule isn't showing up in Airport Utility.  Now What?