You can actually get this old piece of technology to work fairly well with a few easy steps.
- Power off the Airport Extreme and any other router it is connected to for 20 seconds.
- Turn everything back on and wait for lights to stop flashing.
- On your iPhone go to Settings > Wi-Fi. Turn on the setting for Ask to Join Networks. This will make it so the phone always asks you before joining an unknown wireless network.
- Now go to Settings > General > Software Update. You will be prompted to Select a Wireless Network with the Apple Network abcdef as an option. Touch it.
- You should now be connected! There is no need to actually perform the rest of the software update at this time.
I think this works because - one, the software update tool really wants you to download (very large) iOS updates on Wi-Fi to save your data. And two, the Airport extreme for some reason only works if you ask to join the network such that the network selector will pop up. It's a hack, but it works!