Umm, I think the USB port on an Airport Time Capsule was designed only for printer sharing or hard drive sharing on your network. You can not connect to a Mac with a USB cable to access the Time Capsule drive inside an Airport.
The WiFi connection, or a hard-wire Ethernet cable, is the only way to restore data from the internal Time Capsule hard drive in the Airport, unless you take the device apart and physically remove the hard drive and put it in a USB enclosure or use a bare cable adapter.
Restoring all that data over the network will take a long time. I suggest getting an external hard drive to directly back up the old Mac, then connect that backup drive to the new Mac and restore the files from there. When you are done, you could hook that hard drive up to the USB port on the Time Capsule and use that for Time Machine to make a future migration faster, since you would have the option of using a direct USB connection to another computer, if that makes sense. You can connect a new USB drive to the Airport Time Capsule, and copy from the existing backup to the USB connected hard drive. Then you can use that USB backup copy on another computer.
There is also an option of Target Disc Mode, where you connect the two Macs together directly to share the internal drive from one to the other, but you may need some specific cables and adapters depending on what type of ports are available on the two Macs from different time frames.