"Then I should go. Plans must be put in motion. Continuity of civilization has to be considered. I never imagined this day would come."
She talked about putting plans in motion. That doesn't mean that she dreamed up those plans after their homeworld was lost. It is more likely that those plans already existed.
It could be much like the Crucible. Those plans were found and many civilizations of previous cycles worked on that. Our cycle was able to complete it. Even the Intelligence was surprised by it.
An ark could be developed along similar lines. Such an ark could already exist and the asari found it and were able to use it, after some repairs, or, just like the Crucible, the asari were able to finish building it. Probably with the help of the turians and salarians. Again this could have taken place at an earlier time. All they needed to do now was fly it to Andromeda.
It makes sense to keep that a secret. You wouldn't tell the people that you were jumping ship in the middle of a war to save a small minority, right?