Iceland is doomed.
Sorry, but that's the way it is. Granting 2 to 3 months as a lead time to zero Celsius, that is simply not adequate time to prepare. Much has been said about digging shelter, but Iceland simply does not have deep soil which might make "digging" possible. Instead, we're talking about strip mining, and Iceland has no equipment suitable for the job. If we hand-wave that issue away, digging a hole is only the first step. Once you have a hole, you need to erect a structure which will support the overburden which will cover the structure, and that is a major undertaking. At the least, production of reinforced concrete (the best material for large structures of this type) will take weeks to months to organize - and if the refill process has not started shortly after the temperature drops below freezing it will not proceed at all. Frozen dirt is pretty much indistinguishable from solid rock for these purposes.
Deep mining techniques (boring/tunneling machines) won't work, either. On the one hand they are slow, and on the other hand, Iceland doesn't have any of that equipment, either.
But let's say that a certain amount of structure has been accomplished before the Big Freeze. Food production is now an issue, and either "normal" farming or hydroponics will require massive preparation. Keep in mind that, at this point, it has started to snow, and very heavily. This will continue until the oceans are frozen over, although the area of worst snowfall will gradually move south as more and more of the ocean freezes. Due to proximity to the north polar ice cap, things will get bad at exactly the wrong time, and moving equipment will be a nightmare.
And how will farming be lit? It's true that Iceland has a relatively large geothermal generating capacity, about 700 MW, but this is concentrated in two areas of Iceland, one the west and one to the north, and 70% of the power produced is used in aluminum processing, so the distribution lines to the rest of the country are much smaller than is necessary to allow full use of the capacity. Any use of the aluminum power will require construction of new substations, and that is not remotely a 2-3 month enterprise.
So the amount of habitable underground shelter is very limited. How sustainable is it?
The answer is, not. Food production is not sustainable. The geothermal generators will need spare parts and eventual (less than 50 years) replacement. Iceland does not have the industry required to produce such replacements, and there is simply no reason to project the existence of an alternate source.
And just in case some are thinking about fish as a food source: forget it. Within a few weeks to months the sunlight levels over any unfrozen water will be too small to support diatoms, which will cause a cascade of extinctions, with the largest carnivores dying off within a year.