Icing the part of the bottle with air in it slightly lowers the pressure, because the air molecules rattle around less when they’re colder. Thats what causes a little bit of water to boil, until the pressure builds back up.
There’s a much more important effect. The bottle mostly has water vapor, not air, up in the top part of the bottle. Icing it cools it enough so that there the vapor condenses back from a gas to a liquid. That lowers the pressure a lot, causing more boiling. So with the main part of the bottle is very hot and the top part almost as cold as ice, water keeps boiling up from the hot part and condensing at the cold part.
The drip of cold water will gradually cool the hot and stop the boiling. Next time you want Mac n Cheese don’t break out the ICE cubes.