How long would the Earth last if the sun became a black hole?

[halfeye]

Technically true, I suppose, in that you could express the time for the atmosphere to freeze out as some number of hours, but it'd be a large number. Consider that it's quite routine for parts of the Earth's surface to be without sunlight for twelve hours or more, and the atmosphere never freezes out overnight. Nor is heat transfer from the sunlit parts of the Earth relevant, because the Earth rotates way too fast for that.

I'm guessing, but I really do suspect you are mistaken, the thing that would make it quick is that once it starts, it will accelerate. When the temperature drops, the clouds will disappear, and clouds retain some heat, When the pressure drops, air will rush in from elsewhere and that will cause the pressure to drop where the air came from.

The sun pumps a lot of hear into the Earth, currently we're near equilibrium, but there's a lot coming in, and a lot going out. If the in stopped, there'd still be a lot going out, for a short while.

It's not going to happen, so don't worry about it, it's just a discussion on the internet.