How do you calculate the effects of precession on elliptical orbits?

A good starting point would be <insert name of some scientist from long ago> planetary equations of motion. For example, there are Lagrange's planetary equations (sometimes called the Lagrange-Laplace planetary equations), Gauss' planetary equations, Delaunay's planetary equations, Hill's planetary equations, and several more. The common theme amongst these various planetary equations is that they yield the time derivatives of various orbital elements as a function of the partial derivatives of the perturbing force / perturbing potential with respect to some generalized position.

In general, the only words that can describe the result of this process at first is "hot mess." A hot mess did not deter those brilliant minds of old. Via various simplifying assumptions and long term time averaging, they came up with fairly simple descriptions of, for example, $left langle frac{domega}{dt} rightrangle$ (apsidal precession) and $left langle frac{dOmega}{dt} rightrangle$ (planar precession). You can see some of this in the cited 1900 work by Hill below.

While these techniques are old, these planetary equations are still used today. That sometimes you do get a "hot mess" is okay now that we have computers. People are using planetary equations coupled with geometric integration techniques to yield integrators that are fast, accurate, stable, and conserve angular momentum and energy over long spans of time. (Normally, you can't have all of these. You're lucky if you get just two or three.) Another nice feature of these planetary equations is that they let you see features such as resonances that are otherwise obscured by the truly "hot mess" of the cartesian equations of motion.

Selected reference material, sorted by date:

Hill (1900), "On the Extension of Delaunay's Method in the Lunar Theory to the General Problem of Planetary Motion," Transactions of the American Mathematical Society, 1.2:205-242.

Vallado (1997 and later), "Fundamentals of Astrodynamics and Applications", various publishers.
Other than the hole it punches through your wallet, you can't go wrong with this book.

Efroimsky (2002), "Equations for the keplerian elements: hidden symmetry," Institute for Mathematics and its Applications

Efroimsky and Goldreich (2003), "Gauge symmetry of the N-body problem in the Hamilton–Jacobi approach." Journal of Mathematical Physics, 44.12:5958-5977.

Wyatt (2006-2009), Graduate lecture course on planetary systems, Institute of Astronomy, Cambridge.
The results of the Lagrange planetary equations are presented on slide 6.

Ketchum et al. (2013), "Mean Motion Resonances in Exoplanet Systems: An Investigation into Nodding Behavior." The Astrophysical Journal 762.2.