NavList:

 It is relatively easy if you have some kind of clock.
It does not have to be set to any particular time.
The clock is only needed to time a short interval.
You choose a star that culminates around the time
of eclipse. With a sextant you follow this star to determine the moment of culmination (an observation similar to taking the altitude of the Sun at noon). At the moment of culmination, you start the clock. And then time the eclipse. Or, if the culmination happens after the eclipse, you do vise versa. You can also use the Sun, if you can trust your clock for, say up to 12 hours interval.

Ptolemy had a water clock, and a meridian circle to detect a culmination. 

It does not have to be a culmination. Assuming that you already know your latitude, you can take a sight
of any star and find your local time from the formula
sin h=sin(phi)sin(delta)+cos(phi)cos(delta)cos(t),
where h is the altitude, phi is the latitude, delta is the declination, and t is the local hour angle.
You find the local hour angle, and knowing the right ascension  of the star, this gives you the local time. 

If I do not mistake this is called a "time sight".

There is no record that Ptolemy actually used this method, or of any determination of longitude in antiquity using eclipses. But there is a record that Columbus tried an observation of a lunar eclipse in America. His result was very poor: many degrees off. I don't remember how many degrees, but remember reading an article on Columbus astronomical observations. Probably it can be found on Internet.

Alex.