So, Nibiru finally makes it here - we've been waiting long enough.
1) Four years for planning and making changes means the ISS is totally inadequate as a generational ship. There is no reason to believe that the ISS will be any more inherently prepared for such usage than today. I.e., the ISS is dependent upon regular supply runs from earth to replace consumables such as the CO2 scrubbers, food, etc., as well as fuel to keep the ISS from deorbiting.
We don't know how to create a stable ecology good for hundreds of years, much less getting it all figured out and installed on the ISS in time -- esp. given the chaos on the Earth that will be occurring during those four years.
2) The ISS will also become unlivable due to increased solar radiation (or something else). Unless the ISS is somehow given escape velocity, it will also be orbiting at 110 Gm and be expected to incur a 50 C / 90 F temperature rise. Possible to mitigate perhaps in 4 years, but you also need to protect against 37% more frequent solar flare and CMEs (that average a little more than 37% more intense). ISS systems will simply fail over time, e.g. solar panels do not last forever, the ISS cannot possibly manufacture them and you cannot have enough spares of everything to last for hundreds of years. You cannot have an industrial base needed to keep things working. Given all of the chaos, a sudden depressurization event seems more likely than today too.
What is the generational effects of null gravity? Unknown, but it is near certain that a future generation attempting to return to Earth would be unable to function well at 1 gee.
3) Changing to Earth's orbital radius from 150 Gm to 110 Gm is not a temporary change. The momentum transfer from the rogue planet (a.k.a. Nibiru) to Earth is a permanent change unless you have another planet or a second pass to restore the momentum. So, even a generational ISS would not help to repopulate the surface as it will remain inhospitable forever.
The projected average temperatures more likely average about 65 C / 147 F when your consider the difference in solar radiation (1/r^2 law) and blackbody radiation (T^4 law) assuming greenhouse effect remains proportional. Since CO2 emissions should pretty much stop, this may be true. Likely vulcanism effects are contradictory dust (cooling) and sulfur dioxide (warming) so net is hard to predict, but SO2 would persist longer, so perhaps somewhat higher temperatures are likely -- but would seem very unlikely to boil off the oceans. So perhaps Earth is not quite as inhospitable as you suppose.
BTW, living underground does not really solve the temperature problem in the long term as the increased surface temperature continually will seep down until it reaches thermodynamic equilibrium, i.e., same as the average surface temperature. This will take a very long time - hundreds or thousands of years if you are 100 meters or more under the surface.
4) Antartica here we come -- reserve that prime beachfront property now. Of course, guessing where the rising ocean will stop is a challenge. With a 50 C temperature rise, the average temperature in the interior would still be below freezing but the temperature at the perimeter would be well above freezing. There should be a reasonable large habitable zone. Raising crops will still be very challenging due to the long dark winters and weak sunlight in the summers, but some farming would be possible and doubtless quite effective if you have to grow lights in a greenhouse. If you are lucky, thorium cycle nuclear plants will be commercially available by 2037, but failing that, there are doubtless plenty of fossil fuels available on Antarctica - its not likely you are going to care much about man-made global warming. Thorium fuel cycle means you don't have all that complicated Uranium enrichment, etc., but failing that some CANDU reactors might be nice.
Weather patterns will doubtless be interesting and it may in fact get pretty hot during the summer, but people live in hot places today. I would expect it to be livable if not pleasant.
5) Nibiru would be very likely be detected more than 4 years in advance. Our space based detection tech is getting really quite good. You don't say how fast Nibiru is moving. If it is just moving at galactic speed (<100 km/sec), we are going to see it well in advance, automated systems are scanning the skies looking for asteroids, etc. and getting bigger and better. Compared to asteroids, Nibiru is a very large target and would be detected at a distance that would allow for more time considering the great distance it will have to travel before the big event. Consider that Sedna was detected at 86 AU and Nibiru would have to be larger to have the effect described. If Sedna were launched at 100 km/sec it would take 4.1 years to get here. Given the improvements in scanning and the larger size of Nibiru, I would expect that 10 or 20 years is more likely, even given an unusually high speed for Nibiru. I doubt we could make ISS generationally viable with 10 years, but with 20 years I could be more hopeful.
Hyper-velocity rogue planets are very much rarer, but are expected to travel up to 5% of the speed of light. At .05c (15,000 km/sec), we would not have much time, perhaps even less than 1 month.
But I think I have been totally underestimating the ability to detect Nibiru. If the planet is all at warm from the heat of formation or internal nuclear heating from uranium, etc. it will give off noticeable infrared. This means, we will be able to detect it from at least a couple of light years distance. I would also say that this is very likely. Even a hyper-velocity rogue due to hit in 2041 could already be detected by us in 2015. As the nearest discovered rogue, it would be intensely interesting and the race would be on to prevent Nibiru from messing up the Earth. Feel free to post the followup question.
6) No reason to expect governments to cooperate. I would say cooperation is quite unlikely. US and Canada are quite friendly in most ways today, but the northern territory might be prime real-estate. A US invasion of Canada does not seem impossible to me. China invading Siberia would seem very likely as they are not too fond of Russia already. When survival is at stake, the gloves come off. Use of nuclear weapons seems quite likely.
Quibble - A near pass from Nibiru would not leave Earth in a nice near circular orbit. At best you have have a major semi-axis of 150 Gm and a minor semi-axis of 110 Gm. It would be necessary to actually make 2 passes to result in a near circular orbit. The first pass to change orbit to a 150/110 Gm ellipse and a second pass when Earth nears perihelion to change the orbit to a circular one.
Now the Real Problem. Changing the momentum of the Earth in a single or pair of short-term events would be very stressful. What is the gravitational potential of the Earth relative to the Sun? E(potential) = - G * (m1*m2) / r
G = 6.67408 × 10-11 m^3 / (kg * s^2)
Sun mass = 1.988E30 kg
Earth mass = 5.972E24 kg
For e=1.5e11 meters, Ep = -5.28246e33 Joules
For e=1.1e11 meters, Ep = -7.20336e33 Joules
So, earth must lose 1.92909e33 Joules of gravitational potential to achieve the new orbit. How much energy is this really in comparison?
Hiroshima bomb 6.3e13 J
The Tsar Bomba 2.1e17 J
Total annual global energy consumption 5e20 J
Total global fossil fuel reserves 3.9e22 J
Chicxulub impact 5e23 J
Total solar energy impacting Earth per year 5.5e24 J
Total solar output 1 second 3.8e26 J
Rotational energy of earth 2.1e29 J
Total output of sun per day 3.3e31 J
So, total output of sun for 58 days. Or the total solar energy hitting the Earth for 350 million days i.e., about 1 million years worth of solar energy.
I suspect we have a little bigger problem than a new orbit closer to the sun. I.e., the scientists assuring you the Earth would survive were lying to you and hoping you were too lazy or incapable of doing the math. The tidal forces are simply going to be over whelming. Without assumptions and doing the math I can't be sure, but I suspect the only way to transfer this much momentum in a short event duration is to have Nibiru impact the earth. In will also disturb the asteroids belt as well as Kuiper belt and Oort cloud objects. The local neighborhood is going to be unpleasant for a very long time.
Woke up this morning and realized some simple assumptions that make it possible to think about changing the orbit. Assume Nibiru is an Earth twin, this makes the Roche limit equal to 2.5 * Earth/Nibiru radius. Earth radius is 6371 km, so closest possible approach of Nibiru would be 15972 km (center to center) to keep from breaking up the Earth. That is clearly too close, but a useful upper bound. The problem is that Nibiru would have to remain at the Roche limit for 6 days to transfer that much momentum, and of course Nibiru is actually making a fast flyby. So, the proposed momentum transfer is not possible without destroying Earth - actually breaking it into tiny pieces. No planet could impart enough of a change in a single high speed pass to keep from destroying Earth. Minimum speed of Nibiru would be high speed as it must fall toward the sun to approach Earth. Redoing the calculation for the planets grazing each other and I now know for certain that an impact is required to make the required momentum transfer in a single event.
I should also add that exploding Earth into millions of chunks will make for some unpleasant days for anyone trying to inhabit anyplace else in the inner solar system, given the high frequency of big honking meter impacts as well as innumerable smaller impacts.