I'd say it goes like this:
Lara gets preggers and shortly after, some weird earthquake starts happening. Jor-el, a scientist, is unhappy with the way that the council is treating the earthquakes (basically dismissing them), and worried for his families future, he starts to conduct his own research for his own peace of mind.
Within a few months, he's discovered the horrible truth (krypton is doomed) and presented it to the council. But they have dismissed him still.
He then sets about building this prototype, out of his own money and materials, and in his own time. In THAT time, feesibly the entire of Krypton could have been working on a fleet of star ships to get their people to safety if they'd listened to him. If one man can build one ship, surely the planet could have sorted out a fleet of ships.
And then baby Kal-el is born. We have no idea how many months old he is by the time the ship is finished and the planet is doomed.
Seems alright to me