You need to find a proper metal tank radiator for your car, Atlas make one, Loads Ltd make one too. They are very good quality, use a new radiator cap too, and buy a new thermostat valve, also check the condition of the water pump intake pipe.
I would suggest you first do a radiator flush with your old radiator in place, It will clean the rust in the block and head, then install a new Tstat valve and radiator. The sudden heat rise you are seeing is air trapped in the system. You should also check the condition of the small coolant tube that is welded to the valve cover breather pipe. That does corrode and cause air leaks because its a high point in the system.