There's no reason you can't have a multi-generational ship that takes a thousand years to arrive. You don't have to send thousands of people either, it's enough to send a hundred or so, have them populate on the way, and maybe send some frozen sperm & eggs with them to increase genetic diversity.

Another possibility that'll probably become viable within 50-200 years is to not send any humans at all, but to only send a fast & small ship with human sperm & eggs that can start hatching humans once it gets there. Raising the infants could be done with robots and other infrastructure manufactured once we get there.