As Ptolemy describes, for equinoxes, he had a flat bronze ring installed parallel to the equator. When the Sun is one one side of the equator, the corresponding side of the ring is illuminated. The transition moment is the moment of an equinox.
The problems with this method: a) the equinox can happen during a night, or when Sun is obscured by the clouds, b) how precisely you can install this ring, c) how precisely you can detect this moment.
With solstices it is a bit harder. One needs the moment when the Sun attains its maximum declination. This is essentially done by measuring the height of the Sun over the horizon, with the help of a mural quadrant, described by Ptolemy. Then one has the same problems a,b,c, but the main problem of the method is that near a solstice the declination changes slowly, and it is hard to detect the exact moment when it reaches maximum.
An modern analysis of errors of Hipparchus/Ptolemy in these measurements is contained in the thesis of John Philipp Britton, On the quality of solar and lunar observations and parameters in Ptolemy's Almagest, Yale Univ., 1967 (available online).
Remark. All this is not of pure historical interest,
since astronomers tried to determine the actual secular variation of the Earth rotation using ancient observations. This raised the question of actual accuracy of these ancient observations.