I will only answer on the technological side of it, which is the one I know. First of all, instruments to measure time are quite old, as the Egyptians, as far as the second millennium BCE had kind of graduated rules where the shade gave the moment of the day.
Another ancient tool is the gnomon that was used in Greece and the Roman empire, but also in the Chinese world in the second millenium BCE or the beginning of the first.
A bit later we see the apparition of the sundial, as a refined form of the of the gnomon (it needs a bit more knowledge in geometry to work well).
At this time, life was very influenced by natural lighting as well, so hours (at least in the Roman world, somebody could probably elaborate for other regions) were not the same duration at each time of the year. Basically, the night and the day were cut into twelve hours : day hours were longer in the summer and shorter in winter.
For shorter duration (or with uncommon larger systems that were mostly a display of technological power for the powerful of the day), you could also have used a clepsydra, which is a water clock. There is a variety of models, some simply one container with a hole of known size and some kind of graduation to indicate the level of the water still in the container, and far more complex devices with displays and so on. Ancient hourglasses (or variants with other products than sand) are also thought to have existed in some regions as well.
So as a rich Greek plotting, or simply willing to meet a friend or business partner at a convenient time, you would probably have checked the time on your private sundial before heading to the meeting and checking the time on a public sundial en-route.
We even have roman texts complaining of the daily pace dictated by the sundial, as a symbol of the modern, busy life.
EDIT : typo
More can be written, and /u/Vaevix did so. In addition, you might like to read the FAQ sections "Hours, minutes, and seconds" and the next section, "Waking up without alarm clocks". The article titled 'How did people have meetup times before clocks? How did the average person "get up on time" for work? How did they know when to arrive to be "on time" throughout history before clocks?' appears to most closely approach your question.
This is not to discourage discussion. Further questions, data, and debate are always welcome.