There are approximate hand calculations you can do following Kepler's law but one quickly runs into problems that need to be solved numerically. Half-way through this answer I realised this is almost certainly a duplicate of some prior questions so you may find it gets closed by moderators, but I started so I'll finish.
If you want TLEs to be your starting point then the usual assumption is that you need an SGP propagator (a numerical model) to get anywhere. However in the interests of understanding why, here is a bit more explanation:
- Let us start by throwing away some of the information implied by the TLE and say that the orbit is just an ellipse with an eccentricity and a
semi-major axis. We can then solve for position and time anywhere around
that ellipse using geometry. Some bits of this are hand calcs, depending on which way
you approach it.
- The ellipse is itself in motion relative to the Earth turning
underneath it. This is still a hand calc and for some orbits this is still ok to predict roughly where a satellite will be on the next time around.
- The ellipse also does not remain fixed both with respect to either the
inertial Earth, the Sun or the background of fixed stars. There are
some relations for the Earth oblateness (flattening of sphere),
tri-axiality (lumps corresponding to major continents), the gravitational effects of any number of other solar system bodies starting with the moon and the sun, solar radiation pressure and the
Earth's atmosphere. Some of these change the orientation of the ellipse or its shape so that to still call it an ellipse is just a convenient handle. There are hand calc approximations for each of
these but by the time you put it all together you will have made your
own numerical propagator.
- You can do a numerical solution considering i) each orbit at a time, so that each update is an average for that orbit,
ii) a full force model that runs at a smaller time step, or iii)
somewhere between the two. The SGP propagator that works with TLEs
follows approach iii) and is well known to be inacccurate for some purposes but ok for predicting most passes.
Things you can look up:
Accuracy of TLEs
Doing sums
Terminology
