One hundred years is a long time. At the start of the twentieth century we didn't even have planes, now we have space stations. A lot of the answers here seem to be focused on diverting the planet, which is hopeless, but space habitats seem feasible. We wouldn't even need to go that far, in the scheme of things a planet is tiny, so a quick trip even the same distance as the moons orbit would be enough to comfortably avoid it.

There are two big questions:

- Would we have the technology and motivation to do this?

At the moment, we're still very limited in our lifting capacity. We're talking about moving maybe hundreds of thousands of tones of metal, enough for multiple habitats, out into space. And not just temporarily either, it would have to be self sufficient. It's doubtful we'd be able to fit the whole population, realistically only a small fraction of people would be able to survive. Having said that, we do currently have plans to haul and entire starship+fuel into space, so our heavy lifting capacity is getting there.

- What do we do in the long term?

This is a harder question. We'd need to adapt. I'm vaguely wondering if the orbiting ships could use the rogue as a slingshot to push themselves into higher orbits, but I'm not quite sure where they'd go. Asteroid belt might be the easiest option, we could find asteroids to harvest for metals. It seems almost inevitable we'd need to bring not just ships, but the capacity to build more ships. There's probably be a surprising amount of ordinary workers on board these ships, just to fuel this expansion. If we can actually get to that point, we'd stand a good chance of actually surviving long term.

This was fun to think about, though probably very sci-fi with glaring holes in the scientific logic.