Car recently overheated

Advice please. My car recently overheated. On opening the bonnet, there was steam issuing from the area around the overflow on the tank that (I think) feeds the radiator.

As far as I can see, the cap has split, and instead of drawing water into the radiator, it has drawn air in. I'm getting a replacement cap today, but aside from fitting it, will I need to do anything else?

Symptoms so far: Engine overheats after a couple of mins driving. Turning fan on inside the car only produces COLD air. Temp guage inside the car rises after a couple of mins to reflect overheating. The top pipe from the radiator (I think - thick rubber pipe) feels warm / hot after running.

The bottom one feels cold. Feels like there's water in both, but the bottom one may have some air in it too - can't tell. Also, when topping up with water, after the couple of mins driving, the pressure builds up with the boiling water, which gushes back up the inlet pipe and out the overflow.