The warp drive is something that would need a massive breakthrough and a shift in whole understanding of spacetime. In a thousand years? Maybe. What I fear is that according to Fermi, every civilization destroys itself before that point, because it's always weapons. New technology is usually developed as a weapon and only after that, other uses are considered.
Anyway, what I do believe will be possible in next few hundred years are CA/CD generation ships. Once we sort out the effectivity of propulsion (ion engines or a kugelblitz drive) we will be able to build a ship capable of accelerating at 1g up to a decent fraction of the speed of light (80-90%). Such ship would be able to cross the galaxy in about 30-40 years. Well within the lifetime of the crew. Naturally, from the Earth's perspective the journey would take 100.000 years, so no happyend there.

But this, unlike the Alcubiere drive, is something I would believe we are capable of on our current technological level. All except that damn propulsion is doable.