> The numerical accuracy and calculations needed for getting the spacecraft so close to Pluto must be pretty awesome.

You can actually get by with relatively mediocre precision since the craft can be remotely piloted and its trajectory adjusted as needed. NASA was sending corrections to New Horizons just hours before its periapsis to Pluto.

It would be very different if we launched a satellite and then couldn't alter its trajectory in any way (and it would probably impossible to get it to do anything interesting since even the presence of just three bodies can lead to a chaotic system).