The temperature will remain as a constant -40 degrees all around the globe
Let's look at what this will cause:
The New Zealand Subantarctic Islands have a tree line marking the difference between areas in which trees can survive and areas in which they cannot. The difference between survivable and non-survivable areas is the mean annual temperature— above 5 °C (41 °F) is survivable, while below it is not.
-40° is well below that level. Plants will not be able to grow at all. I'm guessing that some seeds would be able to survive the cold, but not indefinitely. My guess is that after a decade the survival rate for even those hardy seeds might be low enough to threaten the existence of all life on Earth, leaving it barren and lifeless even after the winter ends.
The weather is going to heavily affected as well. Wind is caused by differences in atmospheric pressure between two areas. Differences in temperature is one of the main reasons for differences in temperature, so with the planet being a uniform -40 degrees, wind will be gone. Air that cold doesn't hold moisture, either, so once the air has released any moisture it had before the winter began, there won't be any more snow.
This is both bad news and good news—wind power and hydroelectric power are both going to be in trouble (no wind and no rain to fill rivers/reservoirs), but solar power will do well (no new snow or wind to blow snow onto panels).
So how do nations handle this?
Without advance warning (which you can't really have when you don't know what's causing it), there are no advance preparations. As soon as the winter begins, there are two countdown clocks that also begin. The first is the time until food supplies for the general population run out, and the second is the time until existing food supplies for the government's survival program runs out.
Once general food supplies run out, the nation's infrastructure goes down. Obtaining raw resources will become much more difficult. Whatever plan the nation comes up with for survival becomes significantly harder to implement unless it can be finished before that happens.
There's no way for them to save everyone. There's almost 1 billion acres of farmland in the USA alone. It would take a massive effort spanning decades in order to build enough greenhouses and hydroponics facilities in order to replicate that level of production. The nations are going to calculate a realistic number of people that they can save, and then go to work attempting to save that many people.
So what will their Ark of choice be? I think it would be very similar to what people imagine a moon base will look like. It will be well-insulated, so as to minimize loss of heat to the outside. Any areas above-ground or that receive sunlight will be devoted to agriculture, with an emphasis on crops that provide the most nutrition per unit area. The nearby area will be covered with solar panels and skylights (perhaps using mirrors to allow collection from a wider area) to provide electricity and reduce the need for electricity, respectively.
This Ark will not be a single dome with everyone in it; it will be a network of interconnected nodes, where each can function independently of the others. If one fails critically, it can be evacuated and abandoned without threatening the integrity of the others (other than by requiring them to support more people).
One major goal for the nation will be to save at least a minimally viable population. If they have fewer than around 4000 genetically-distinct individuals, they'll need to work harder to make sure that the population can survive long term. I believe that you can go down to a couple hundred people and still survive, but only if it is for a single generation and you implement a breeding program.
Each node of the Ark will also include a storage room for when the winter ends. This will have plenty of seeds, equipment, and instructions to help the survivors to rebuild after the winter ends.
So the nation just needs to build as many nodes in the Ark as it can before the nation's infrastructure falls apart (the first countdown) and then make sure they are self-sustaining before pre-winter supplies run out (the second countdown).