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FIrst off, remember that the times of "sunrise" and "sunset" are a gamble, as refraction is highly variable near the horizon.

The typical definition of sunset today is when the sun has an altitude of -50', which includes an assumed refraction of -34' and an average semidiameter of the sun of 16'.  So, from a CelNav point of view, you could find a line of position at an assumed time (say, from sunset tables) and then determine the difference in longitude along your line of latitude, converting it to time.  If you're looking for a strictly numerical method, you'd basically be doing the same thing, using iteration to find your goal.

As for "exact times" of the equinox, at high precisions (well beyond what a handheld sextant can measure), the terms "equator" and "ecliptic" (which defines the Aries point) mean different things to different people.  For starters, modern astronomers don't define the ecliptic by the center of the earth, but by the center of the earth-moon barycenter; the moon's gravity pulling on the earth causes it to bob above and below the "plane" of the barycenter's orbit.  The end result is the sun typically has a non-zero declination when its SHA is exactly 0° or 180°.  Further, the poles (and with them the equator) "wiggle" a bit, both with respect to the stars and to the surface of the earth, so you'll need to decide if you're going to use the long-term average position of the equator or its position at that instant.