Define what you mean by "this cable". What is the make / model of the USB ethernet adapter. Support depends on the chipset of the device. I am not sure what you mean by:
This cable is intended to be USB to RJ45 not the other way around. Apparently USB can connect to RJ45 but RJ45 cannot send to USB.)
I have an older Linksys USB100M adapter which uses the rtl8150 chipset. A USB ethernet adapter is typically a dongle with or without a short USB cable on one end and a RJ45 socket on the other. One must then attach a CAT5 cable between the adapter and in your case the router. Hopefully linux will automatically detect the adapter and load the necessary module. Look at the output of the dmesg lsmod and lusb commands to see if the device is being recognized. If all of that appears to be good then look at the output of the
/sbin/ifconfig -a command. You should see a eth0 device. Post the results.
USB ethernet adatper examples: