Throughout history, people have not all thought of time in the same way. Or planned for it.1 Or talked about it.
In part because they have not had the same ways to measure it.
The first and most obvious way to keep time is to watch the skies. Until modern times, all your characters should be more acutely aware of, and able to read, the sky far more than modern ones, even many astronomers. They are familiar with landmarks that give directions and seasons, and can read by the slant of the sun how late in the day it is, by the moon how many days it has been, by the progress of the stars how late at night it is (and how late in the year as well).
This will make their time keeping much more lackadaisical than any you are used to. Any time vaguely near solar zenith is noon, and you would have be there very late indeed before anyone would tolerate your being declared too late. Hours are improbable. Minutes and seconds are incomprehensible. In such an era, all your characters will think, and talk, in this manner. And your narrator needs to talk in the same way, or the result will be comic.
(The characters would also be very aware of the sun, the moon, and the stars because they were the only free lighting sources. Gathering wood and other fuel for fires was a major issue. You would see by sunlight, moonlight, and starlight whenever you could.)
But what if the characters want more precision?
Sometimes, indeed frequently, you were simply stuck, but various techniques have been used.
For short periods, reciting things such as prayers in a fixed cadence can time recipes or test how hot a fire is (by how long you can endure having your hand a fixed distance from it). Many rhymes, pronounced over some medical brew, probably had value as a timing device rather than a spell, whatever the brewer thought. Then, such things give only relative times.
For absolute times, sundials are ancient, and perennial. Much easier to glance at a shadow than at the sun. They do need rather more astronomical knowledge than may be apparent at a glance. (Wizards live in towers to make it easier to observe the stars and work out calendars and such things. And wore clothes with stars and moons as a symbol of this very important work.) Then they need to be arranged in place, so they are only for a small area.
Sundials do let you talk of hours, but those will vary with the season, which will be worse the further from the equator you are. Nightly hours shorten as daylight ones grow, and vice versa; the night might not have hours, but four or five watches for the sentinels. After all, even if you have them, sundials do not work at night. Worse, they do not work on cloudy days.
The prevalence of sundials still was heavy in some places -- less so in others. In ancient Rome, a stock comic character was a man lamenting how, in the city, the sundials ruthlessly chopped his days into hours.
Candles, if you have them, and a hurricane glass or the like, so that you can see its progress, and its progress is likely to be regular without the influence of drafts. Medieval times would use horn, cut very thin, to form the lantern. The real problem was that such a clock required a candle, which was not cheap. (Also, characters would talk of distances on the candles, probably in finger's widths.)
Water clocks are calibrated to measure time by how much water is left in one container, or added to another, pouring from one container to another. This gives only relative times, but it has its uses. Aristotle discussed how they were used to time plays in competition. (To contrast an artificial limit of story length to the natural, aesthetic limit of a story.)
These evolved into the hourglass by the late Middle Ages. Much more reliable, endlessly reusable, and best of all, cheap. They gave only relative times and except for the time they were engineered for, only vague times. Still they were cheaper than clockwork.
Clockwork was also coming into play in this time. It was not wildly accurate, but it allowed many things. Monastic orders could use them to time the Midnight Office, for instance. They also influenced the measurement of hours, because it was much easier to keep a clock in order if you kept fixed hours instead of letting them shift with the seasons.
This era also had the custom of ringing the church bells for the hours of prayer. These would come every three hours -- more or less, depending on whether it dictated by the sky, or a sundial, or a clock -- and would spread the news of the time far and wide.
Minutes also came into use in medieval times. Astronomers used them, and seconds, thirds, and fourths, which were not in common use. Not until the hairspring was invented, in early modern times, could clockwork grow accurate to minutes.
Therefore Shakespeare did write
See the minutes, how they run,
How many make the hour full complete;
How many hours bring about the day;
How many days will finish up the year
How many years a mortal man may live.
but his characters do not speak of seconds. Yours should not, either. Moments. Breaths, heartbeats, jiggers, or any other irregular unit of time, if necessary.
Because some flowers close and open during the day, late in this era also has the whimsical notion of a flower clock, whereby you can see the time at a glance by which flowers are open. This is vaguer than even tracking the sun, and remains merely a whimsy. (Unless you use magic to boost it.)
Clockwork became king in this era. Its progress was such that the regular hours, which have their advantages and which clockwork works better for, took over from the sky. Apparent Solar Noon, when the sun is actually at zenith, is, at times during the year, more than fifteen minutes out from Solar Mean Noon. Obviously in different directions, so it averages out. Only Solar Mean Noon will give you 24-hour days with fixed hours.
The city of New Haven had a contretemps when two public clocks used the different noons, and much indignation that when the sun was at zenith was God's noon, and none of this "mean" nonsense. It got little play, though, because clockmakers vastly preferred the easier mean, and very few other people knew of the issue even in theory, since the idea of noon was so vague.
Not so little known was the later issue of railroad time. How to calibrate all those stops -- and just to add to the issue in Great Britain, some railways used Greenwich Mean Time while others continued to use the local Solar Mean Time. Other countries adopted it, or time zones, more abruptly, but it was never feasible to force everyone to change. A judge once ruled that a man was too late by the court clock, and therefore that he was in time by the railway clock was irrelevant.
Part of the issue was that the telegraph meant you could get your time, far more precisely, from an astronomical observatory, but of course, that would mean standardized time over the region it served. It also meant that meteorology was feasible as long as the reports used a common time scheme, so that progress of weather events could be plotted.
The people who were going by the sun itself got more annoyed at this one. It was the sun, not the railroad, that should dictate time, real time, God's time. (The other influences on standardization were less obvious to the people, so it's railroad time.)
Later, when daylight savings was introduced -- to save energy, for the benefit to the factories, much to the farmers' detriment despite stories to the contrary -- that, too, raised much chagrin, in part because it was tampering with God's time, the erstwhile railroad time.
Part, of course, because with the advances of clockwork, clocks had gotten quite cheap and rather small. Most people had at least one in their homes, and many carried one about. The change has continued on the path. Asking people for the time is unusual nowadays, because not only do we have time to a degree of accuracy greater than ever known, it is known to more people.
Such are the changes of time over time.
More on that here
Taking Time
Stories take time. They unfold over hours, days, week, months, seasons, years. . . centuries and ages if the story gets frisky (and they have been known to do so).
I kept all modern measurements out of my fantasy world, so I had to improvise using the sort of ideas you expressed here. Sometimes it made it quite difficult to express what I wanted to show in the story. But after doing it for years, it got easier. I took an astronomy class in college which helped me see how to use the phases of the moon to tell time.
I get the idea from this that one of the first things someone from a more low-tech world would notice upon arriving in our world (or any other) would be that the night sky was vastly different. I mean, assuming that they don't get dropped down in the middle of the Bronx or the like.