Backing up to an Airport Timecapsule over non-Airport WiFi
The answer is Yes, it can do that but needs to be set up first.

Here are detailed instructions.

In summary:

Connect your Mac to TC via Ethernet to set it up.

Now Disable the wireless functions of the Time Capsule and, Set your Time Capsule to Bridge Mode.--Update the configuration.

Next: use an Ethernet cable to connect one of your router’s LAN ports to your Time Capsule’s WAN port.