This gets into the subject of "chronicity". In pre-industrial societies the idea of time being divided up per day in very precise increments, and the idea of the primacy of appointments at specific times is alien. This is often called "island time" or "polychronic time". Someone who lives with that concept of time generally doesn't prioritize appointments at specific times. In an island time culture an appointment is generally an "around this time" and "assuming it doesn't interfere with whatever you're doing at the time" sort of thing. So the person might get around to it later than planned or not do it at all if they get caught up doing something else.

To those of us who live with "monochronic" time, this behavior seems "flaky" but it's really just about a different values system. Indeed, in a lot of ways it's bizarre to prioritize precise appointments with strangers over other life priorities (such as spending time with loved ones and friends). If you look at the different priorities of folks who have a monochronic versus polychronic sense of time you'll notice that monochronic time people have a tendency to make short-term practical relationships while polychronic time people have a tendency to make long-term familial relationships (and if that doesn't make you feel bad about living in the industrialized world, I'm not sure what would). But it does have some advantages in a mechanized world full of factories and railroads. Buses, trains, and planes run on schedules, as do factories and banks. Working in an industrialized world often means adhering to a fixed schedule, and often means doing errands and other tasks within tightly defined time windows (job interviews, going to the bank, taking the bus, etc.)

Before clocks the vast majority of folks were farmers, and most people had a polychronic conception of time. If you're a farmer you simply don't have a schedule full of appointments. You probably have a ton of work to do, but by and large it's a matter of doing one thing then moving on to the next, then the next, etc. Without factories that run 24/7 and need to be manned in shifts, most work would simply start in the morning and end whenever it was done, and most people lived very close to their work.

People still made appointments, but the typical appointment would be between friends and not especially time sensitive because everyone involved would have a polychronic conception of time.

Note that even when clocks became commonplace it took a long time for time to be standardized from city to city. The two major factors in bringing about standardized time were factories and trains. Running trains by a schedule is the only way to do so efficiently and predictably, which is also true of factories. When you have steam or electricity driving expensive equipment you have the ability to run it 24/7, and it's generally vastly more cost effective to do so, even though it means bringing in people to work during nights and evenings. But those things also mean that people need to start living their lives by a schedule. Having an appointment in a pre-industrial setting would be a rare thing, and also rarely consequential for your life, but when you buy a train ticket making an appointment becomes a big deal, and when your job requires showing up to appointments nearly every day of your life, then the structure of your life changes.

In short, before clocks people didn't really need clocks. And clocks began to be more commonplace as people's lives and people's conception of time began changing due to industrialization.

Edit: I'm going to add some additional material here that was originally going to be in a reply.

I should point out that there's a lot more to the story here than mere polychronicity versus monochronicity and industrial versus pre-industrial societies. Even in the medieval era it was fairly common, in Europe at least, for towns to mark the time throughout the day and night either with bells or musicians. Towns would also typically have watches (and thus "watchmen") set up throughout the night. It was common for a town to have a wall and/or gates and to close the gates at night, disallowing most traffic in or out. Knowing the hour would be useful for planning activities but it's not really known, as far as I'm aware, how common specific appointments were in the medieval era among city dwellers, or how diligent folks were in being on time. Even among occupations that required very regimented schedules to the day (such as sailors) it's questionable to what degree that instills a monochronic sense of time. Even more so, it's questionable how much it instills a cultural valuation of the primacy of monchronic time.

So as a coda to the previous answer, if you did happen to live in a city and you decided to make an appointment there were ways to keep such an appointment (by relying on the tolling of hours from a bell or whatever) but we don't really know how common that was. One thing we can look at is people's opinions about the pace of society. A monochronic society seems a lot more "fast paced" and pressured than a polychronic one, so a common complaint of a monochronic society is that it is always in a hurry. There's a noticeable uptick in such criticisms during the mid 19th century as industrialization comes into full swing, which might be an indication that monochronicity wasn't very common at all before then.

So far, not one mention of the sundial. It's hard to imagine that ancient civilizations, hierarchically organized, would not (at least near the top) have meetings occurring serially at specific times throughout the day. Pharaoh, for example, whose time was precious, needed to meet with "managers" throughout the day, who themselves had to meet with subordinates to prepare for these meetings. These people were obviously highly organized in other ways, judging from their works.

Is there no evidence of monochronic time and time management among the busy, ruling classes in the ancient world?

Before the mechanical clock, the main method of telling time was the position of the sun in the sky. Timekeeping devices such as obelisks, sundials, water clocks, hourglasses, and candle clocks were also used.

People may not have had personal access to such devices but communities used bells and criers to signify the passing of the hour, or even just community events such as religious observances, holidays, curfews, or the arrival of important visitors.

I wrote a smaller paper about the problems factory owners had to face in the 19. Century. And one big problem was that people always came late (or didn't even bother to show up), because they weren't used to a rigid schedule/time frame yet. Before the industrialization, life was usually lazy - one got up and did things when necessary, and stopped doing them when they thought it was done or more work wouldn't be needed anymore.

During the industrialization people had to quickly adapt to the rigid time schedule which was needed in a factory (because of shift changes etc). This was harder for people which knew the life before that and thus weren't used to this sort of time measurement. This is by the way also a reason, why bells were installed on many factories - most workers in the early 19th century didn't own a clock yet.

Being late and skipping work in factories during the 19th century was usually severely punished with either a high penatly (in Switzeland the penalty was around a halfday-earning, which could be devastating), an immediate dismission, compensations (because the worker could be excluded of the factory for a day and the owner hired a fill-in worker, the late worker then had to pay compensations) or even all of the above.

The amount of working time also exploded in the 19th century aswell (nearly doubled the amount, check link below).

This is for Switzerland and Germany, but I guess it was very similar in ither European places aswell.

Sources:

If you want to know more about workweeks and hours worked, also about the 'lazy life' in medieval times (compared to modern ones) the following link is nice:

http://groups.csail.mit.edu/mac/users/rauch/worktime/hours_workweek.html

Also Moishe Postones work "Time, Labour and Social Domination" is very good.

I also got many (german) sources about the disciplinary aspect of the industrialization:

Bernhard Weidner: 'Sozialdisziplinierung in der Industrialisierung', Bernd Flohr: ' Arbeiter nach Mass' or Jacqueline Janser 'Im Takt der Maschinen'. Of course Erich Gruner also picks up this question in his very good work 'Die Arbeiter in der Schweiz im 19. Jahrhundert'.

Sorry for my lack of good English and formation, im not english native and on mobile.