Did ancient cultures use different units of times?

I can only answer for old Japanese method of counting time during Edo times, I'll stick to that.

So the answer would be that there was a concept of hours, but the length of those hours was not constant and depending on the season. A day was divided into 12 "hours", associated with chinese Zodiac signs names and a number. The numbers would go twice from 9 to 4: 9 (middle of the night), 8, 7, 6, 5, 4, 9 (middle of the day), 8, 7, 6, 5, 4.

The middle of the "morning 6th hour" was sunrise, the middle of the evening sixth hour was sunset. Middle of a 9th hour was either midnight or noon.

Quick consequence, depending on the season, you'd have shorter "night hours" with longer "day hours" (the length was changed every two weeks or so). For instance workplaces would have a clear start/finish time (e.g. beginning on morning 7 til evening 7), so as the hours lengths were varying depending on the season, the duration of the workday was also changing.

There were a few bell towers around the city in Edo, that would ring the time.

Each hour was divided into 4 quarters, but not beyond that (so that does not go down to a "minute" to determine a specific time of the day). As an "hour" in Edo was roughly 2 of modern hours, the granularity of time keeping was around 30 minutes, but not more than that.

Towards the middle/end of Edo period (XVIII/XIXth century), there were a handful of mechanical clocks built based on this system (those clocks are called "Wadokei"), as one can guess things can get very tricky with varying hours length. There are a couple of those impressive mechanisms kept into museums in Tokyo.

The move from varying length to fixed length was enacted in 1873.