Those ships are also large enough that the walls can be very thick, preferably packed with soil for growing plants, in order to minimize exposure to cosmic rays.
Such a ship would most likely have to be self-sustaining, i.e. all solids and liquids (and I mean ALL) being recycled and reused indefinitely for very long journeys.
Once you have such a ship, you don't need it to go anywhere in particular. It can just hang about in space for eons.
It would be best, however, if it had a place to go so that some generations hence the remaining humans would be able to live on another planet rather than trapped in the cylinder.
If we spend most of our money now on probes, then we are more likely to find such a place for the cylinder to head towards.
I also wonder if there is anything that could completely destroy a group of people in a sufficiently hardened and outfitted underground bunker. With the advent of VR technology, we could have about 100k people dispersed into various bunkers around the world with sufficient food, water, energy, and entertainment to keep them physically and psychically happy for decades to come.
I would rather be a Spacer on Aurora than to survive in a dystopian Cave of Steal.
The key is ‘sufficiently’ and if there is no warning then no bunker protection will help - many asteroids come in from behind the sun and cannot be seen until they are to close to do anything about if they were on an intersect course. No enemy who is about to launch will give warning. So in either case bunkers will do you no good unless you live there permanently.
Easiest thing is to hollow out an large asteroid like Ceres or Eros and install engines (name your brand), supplies, etc and off you go. Assuming that some asteroids are not already hollowed like the captured moon of Mars Phobos is thought to be ... Then there is Iapetus ...
“Spend most of our money,” there you go again. It is not about spending money. Just throw that out. If you go there, then you are not serious, as there is always a reason to spend less and less until the thing is canceled ... just ask the Russians about their manned lunar program, or even about some of their current plans.