3-body problem implies crazy perturbations and chaos theory and invoking of pendulum analogies, and all that's true, but due to the magnitudes of difference between human timescales and cosmological, we know the eclipses, occults, and all other matter of quirky alignment and other miscellaneous minute for millennia to come.
Would I be able to make the same prediction using only math and whatever instruments could measure the positions of the sun and moon that accurately? What instruments and equations do I use?
Once you have all the data you find all the times when the Moon crosses the ecliptic plane (about once every 14 days), which are the only times when eclipses can happen. You discard the times when the Moon will not be aligned with the Earth and the Sun. You will be left with a handful of times per year when an eclipse is possible. If the eclipse is solar you compute the orientation of the Earth to see where it will be visible from. It's all trigonometry and linear algebra.
The Voyager spacecraft, when being gravity assisted by Jupiter, stole enough energy from it, that in 5 billion years, Jupiter will be 2 millimeters behind in its orbit around the sun.
But you did make me think of this: https://www.eso.org/public/videos/eso1825e/