I assume the spacecraft carried navigation sensors to let it control its route in a closed loop. That's how you can go however far you want without requiring unbounded precision in your initial calculations.
> See the "Guidance and Control" and "Communications" section of the NH
> Spacecraft Systems page for a detailed answer.
>
> The short version is that it uses a combination of star trackers and IMUs
> (Inertial Measurement Units). The star trackers analyze pictures of the
> surrounding star field to determine how it is pointing instantaneously, and
> the IMUs track how it is rotating in between each of those instants. This
> determines the attitude (which way it is pointing).
>
> For position determination, "ranging" tones are sent from the earth and
> echoed back by the craft. This combined with the angle that the dish is
> pointing at to get the strongest signal tells the operators where the craft
> is in space. This information is fed back to the craft, which has an on-
> board physics simulation, and predicts where it will be until the next
> ranging event.
>
> Now, you might have noticed that I didn't mention Pluto once. That is
> because this system (minus the exact details) is used by pretty much every
> spacecraft, from those around Earth, to New Horizons, and beyond.
