A more likely scenario is that this ship was a special project, designed and under construction since Sovereign's attack by some secret section of the government. In that case, the timescale for designing and constructing the ship becomes more believable.
That possibility hasn't been ruled out, has it?
Possibly. The thought had occurred to me too. However, it's still not much more lead time. I forget the exact timescale but wasn't Sovereign's attack about 3 years before the invasion? 3 years to crack intergalactic travel using Sovereign's reconstructed drive still seems a stretch. And it's not even certain that Reapers like Sovereign would be able to get all the way to another galaxy. We don't know how far they had to travel from their base in darkspace.
Such a project would also be very difficult to keep secret, especially from people like Cerberus or the Shadow Broker. Liara should have been able to find out about it, and it would have been reasonable to expect her to mention it to Shepard when talking about her own time capsules.
I continue to wonder whether Bioware really planned in advance to make a fourth game. They didn't seem to lay much groundwork for it in the last trilogy.