I think everyone is collectively ignoring that I'm specifically talking about accelerating the craft from earth, not using an ion drive or similar form of propulsion on-board. As that's what is implied by the comment that I responded to: "If you could shoot it out at close to speed of light"
If you can apply small force over a long time, that will get you up to speed, too.
Someone did the math in the thread, and suggested that a constant 1g of acceleration would get you to the centre of the galaxy in 20 years (as measured by the clocks traveling on your spaceships). 1g of acceleration for 200kg is about 1962 Newton.
(This back of the envelope calculation assumes you have eg someone fire a laser at your ship to give you the energy you need. If you need to bring your own fuel, the rocket equation increases the total mass needed. But the same principle still applies: something like an ion drive has very little force, even if the top speed it can reach can be enormous.)