1) Preparation. 2) The event. 3) End of days.
On a 10-year timescale, the biggest thing we need to prepare for is the fact that almost the entirety of our species is going to die in the immediate aftermath of the
event — how will that impact wider politics and human interaction? Like, in an ideal world, there would be a concerted global effort and we would centralize our collective energies, but given how exclusionary the erstwhile elites tend to be, the countries and people that are the most disfranchised and
need the most help from a technological/preparatory standpoint will most likely receive the
least amount of help — and they might decide to nuke others as a parting feck you...instead of going down with a whimper (provided the elites don't nuke them first). I'd personally much rather trust private efforts than “Save Humanity” brigades fostered by governments, but that private clubs will be very exclusive and limited to the wealthiest and most influential. All things considered, I don't think we can prepare for that event, really — the scale of even our greatest feats is incredibly small. We might be able to prepare for a few thousand people at the max by going deep and leeching off geothermal energy (mostly politicians and businessmen — and engineers/geologists/doctors to keep them alive).
As for the event itself and the end-of-days, how long can we survive underground, and what quality of life will the survivors have, and can they lead a
rejuvenation? If we're sufficiently close to the other planet, we'll have to worry about crazy geological activity via tidal heating as well — how will the survivors navigate underground life in that hellish situation when we can't really predict the post-event volcanology of Earth?
Hmm...I actually think that instead of trying to salvage a lost cause on Earth, humanity would be better served migrating to Mars in the hundreds and thousands, if possible: negligible O2 and fairly low temperatures and not a whole lot of protection from radiation compared to current Earth, but it at least have a Sun and a somewhat stable environment compared with grim future that “Rogue Earth” is in for. Given enough time, I reckon re-population on Mars is more realistic than medium to long term survival on an ejected Earth.
I wonder if we should actually put a solar blocker on the L1 point between the Sun and Earth, and try to actually freeze our oceans prior to meeting the rogue planet? We'll be experiencing this eternal winter soon enough anyway.
Impossible, in my opinion — the L1 point is further away from the Earth than even the Moon...
You would need a proportionately larger (relative to the moon which casts an umbral shadow of just ~250 km) reflector/blocker to influence the oceans to a noticeable degree (
https://rechneronline.de/sehwinkel/angular-diameter.php). I doubt we could build something of that magnitude (and successfully put it into orbit) in a century. Even if we cleverly place a lens with a short focal length at the L1 point or perhaps a diffraction apparatus, the whole structure will have to be hundreds of miles across (again not very realistic and more in the realm of science fiction at our current level).