The time interval formula is all that is needed to compute the times of other events if one has a known position and time for the Sun like Ptolemy's autumnal equinox.
Did Ptolemy a table of time differences from the time of perigee that allowed him to compute the time for passage between two longitudes? Ptolemy's anomaly table may be such a device. Ptolemy gave the angular position of perigee so one can compute the angular distance from perigee for a given longitude. Using the anomaly table in reverse one can determine the corresponding mean motion and the time interval from perigee. Doing this for two longitudes allows one to determine the time it would take to go from one to the other. So if the date and time of one longitude is known one can compute the date and time of another.
Supplemental (Feb 18): The Julian dates assume UT which is Greenwich Mean Time but the latitude of Alexandria is about 30° E so it is 1/12th of a day ahead of UT and this means that one needs to subtract 0.083 from the times above to correct them.
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