Hi!

The connection on the back of the laptop is a phone type. Is there a cable
that can bridge the gap and allow me to use this laptop via cable?

That connector is for use with a telephone line only. It connects to the
system's internal modem. The internal modem only "knows" how to function
with a telephone line. The possibility of damaging something is unlikely,
but what you want to do won't work.

If you have a wireless card, you could invest in a second router and use
that. You might find one in good working order on eBay or at a garage sale
if you don't want to buy new.

William