If he wants you to do this, explain that you will buy a basic $10-20 unmanaged port extender (switch) and then take one plug out of the old router, connect to the new switch, and plug the cable you disconnected to the new one. You can always reverse those steps and reboot things if needed, but I wouldn’t expect this causing any problems given how often I’ve changed ports for things without any need to reboot.

The key here is making sure it’s an unmanaged switch, you don’t want a router doing Network Address Translation behind another router or you’ll have double NAT which breaks certain port-dependent services unless ports are forwarded on both routers, but instead a simple switch doing Layer 2. If you have a spare router, many can be configured in the settings to act as a dumb switch as well, but with an 8 port Netgear switch being $17 that’s the easiest solution.