Going 99.9999% of the speed of light, to go forward 2000 years would take almost 9 years of continuous near light speed travel. We don't have the technology to travel at even a fraction of that speed. Even if we did have the technology to travel that fast, we don't have the technology to maneuver, and avoid things at that speed. The best confirmed, tested technology we have is combustion propulsion.
Even if you did have a nuclear reactor on board, what would it power? You realize nuclear energy is just electricity, right? It just produces a lot of electricity in a relatively small area.
Aircraft carries and subs have nuclear reactors, but they power propellers. Are you suggesting we make space submarines and fit them with space propellers?