I've been thinking about this, and I have a theory as to how the ship itself came into being.
I think the Asari had been building this for quite some time, possibly centuries
before ME1. My theory is that they were intending to eventually go to Andromeda to get that first foothold into a new area. This would have helped establish some superiority/seniority in yet another galaxy. I could see them doing this in order to possibly establish some route between the two galaxies.
From what we've seen so far, it reminds me of a Collector's ship as well, in the sense that this is meant to hold millions of people. The Citadel itself would be a good inspiration, as it was a ship/structure built to house a lot of people. I think the Ark was originally meant for the Asari for their own purposes. To be the first to contact new races; the first to discover new resources; they could set up the government they way they saw fit, ect. They are brilliant and arrogant, so it made sense in my mind.
Now, I think they left around Thessia's Fall. The Asari could no longer bring along their own population as a sole means of establishing a new colony (with most of their race involved in the fighting directly, or were a casualty,) so they worked with the Council members to start quietly pulling some of their qualified citizens and military from various Council races (including humans) away from the front lines. As far as to whether or not they saw the Crucible fire I think really depends upon if they went into cryogenic sleep right away or not.
I'm not saying I'm right or wrong about this, but I tried to think of a good explanation that didn't require tons of mental gymnastics to get to. It would make zero sense if this ship was built just due to the Reaper War.
Finally, when the Asari Councillor says, "I never thought this day would come," I was reminded about something many (if not most) civilizations share in common. Many religions and cultures have stories involving the end of the world, or the end of all life as we know it. Given that the Asari's beliefs came from the Protheans, there could have been a warning passed down to watch for the End of the World. Over time, the message became unclear (like the game Telephone,) and as the Asari became more scienfitically minded, they probably dismissed those stories as relics from a more primitive era.
This is all in my opinion, and stuff I've been thinking about while I wait for the final patch for DA:I to come out. I'm not strong enough in science or math to offer any opinions about
how the ship would travel. I'm only thinking of the story side. Speculation is fun.