There is already a great answer by u/Celebreth here ( https://www.reddit.com/r/AskHistorians/comments/69knux/time_keeping_in_ancient_rome/) on Ancient Greece - I will try to do the same for Rome.

In Ancient Rome, things were somewhat complicated. While everyone could more-or-less agree on what time it was right now, your idea of time shifted depending on where in the world and when in the year you were.

Originally, things were imprecise. At least in the city of Rome itself, time seems to have been reckoned originally by looking at where the sun was in relation to certain landmarks - the Roman writer Pliny, around AD 79, records that a herald would be dispatched in the Forum to announce when it was noon, based on seeing the sun 'between the rostra and the Graecostasis'. This exact measurement cannot have been particularly ancient, however, since the Graecostasis was only built around 160 BC.

In the middle Republic, the system of 'hours' was developed, dividing the day into twenty-four blocks of time, twelve between dawn and dusk and twelve between dusk and dawn. The first hour after dawn, therefore, was prima hora - 'the first hour' - or sometimes prima hora diei - 'the first hour of the day', and so on for eleven more. The first hour after dusk was similarly prima hora, usually called prima hora noctis ('the first hour of the night') - though as we shall see in a minute, this was rarely necessary.

There is an obvious problem with always marking the period between dusk and dawn as twelve hours, which is that the length of a day changes sharply from season to season - as such, an 'hour' could be anything between 40 minutes and 1 hour 20 in our modern system, and would be different in the day compared with the night.

The consul Valerius Messalla brought a solarium - a Greek sundial, now re-named in Latin - back from war in Sicily in 264 BC, but it cannot have been calibrated for Rome's latitude and at any rate would not have worked with the Roman system of reckoning hours. There were quite a few of these around the city - Augustus famously built a tremendous one on the Campus Martius in 9 BC, whose gnomon (the bit that sticks up and casts a shadow) is now in the Piazza di Montecitorio. Others can be found around the city and Italian towns, often on other buildings - in a fictional text called the Satyricon from around AD 60, the aristocratic author Petronius has the boorish character Trimalchio plan to have one put on his tomb, to ensure that everyone would look at his name written beneath it.

From the 2nd century BC, solaria could be used to calibrate water-clocks - clepsydrae, a name which gives away the Greek origin of the technology, was Latinised into horologium - which had the advantage of working when the sun could not be seen. The basic idea was to make a cylindrical 'hourglass' filled with water, fill it up in at dawn, mark how far the water level had fallen by dusk, and interpolate. This could be done at various times of the year - Carcopino says that they would mark twelve vertical lines, one for each month, and then mark a set of horizontal lines on each one as above. I am not however sure of his primary source for this and his writing sounds like he is working out how they 'must have done it' rather than referring to anything empirical. At any rate, we know that it was never an exact science - Seneca, around AD 55, suggests that 'it is easier to find two philosophers who agree than two water-clocks'.

At night, things were more usually done by 'watches', timing by the four rotations of the city or camp watch, each marked by a trumpeter. So when sources talk about things happening at night, they much more usually talk about which 'watch' it was (and in other words, only give the time accurate to about three hours) than the 'hour' - because the solaria would be useless and few people had a horologium (a retainer of our fictional friend Trimalchio picks it out as a status-symbol that will wow his upper-class guests).

Things got even more complicated if you ever travelled - I have seen something somewhere about how people noticed that you could travel West 'faster' than you could travel East, in that you might set off going West at the 'first hour' and arrive to be told it had only just gone the the 'second hour', but then turn around and find that the 'third hour' had been some time ago when you arrived - of course, the problem was that the sun rose later in the western town than it did in the East. However, I cannot for the life of me find the source.

One imagines that Romans would develop a keen sense of time, much like many people who rely on keeping regular time-limits do today - watch a chef or teacher at work to see that. For an analogue, you can think of the Guugu Yimithirr in Australia, whose language traditionally has no egocentric directions - only 'north', 'south', 'east' and 'west', and so who develop a keen sense of direction without necessarily being able to articulate how they know which way is 'north'.

Sources

Pliny's recollection of sun-watching in the Roman Forum is recorded in his Natural History at 7.60. Seneca's quip about water-clocks is in the satirical work Apocolocyntosis

On Roman 'clocks', see the entry under 'time' in Matthew Benson (2002) Encyclopedia of the Roman Empire

There is also a lot of useful information in Jerome Carcopino's 1936 Daily Life in Ancient Rome - dated, yes, but this sort of 'nitty-gritty' factual scholarship largely escaped the major overhauling that came with the theoretical shift in Classical scholarship, and has not been of particular interest to many scholars since.

On the Guugu Yimithirr, see Guy Deutscher's 2011 'Through the Language Glass' - though it is worth pointing out that their system has all but died out in the face of modernisation.