Uploading C program to the ARM core for execution through ethernet!
You would need either


an ethernet-to-jtag bridge (hardware and software): this keeps you current interface intact, including debugging. I have no idea whether this exists for your setup.


or


an ethernet bootloader running on the chip + development environment settings that can work with a bootloader + a way to reset the chip (press the switch?) + a way to let the bootloader know that it has to bootload (instead of running the existing application)(switch? jumper?)


PS I don't know which 'obvious' reason you have for wanting to use ethernet instead of jtag.