"How would" is hypothetical and I can't delve into that here. "How did they?" That is answerable, but apparently not to satisfaction. We do not know for sure. But as far as we understand the texts, it seems to have been not very precise in these very early times.
Take note that "Sumerians" is referring to a time period really early. With texts ranging from circa 3000 BCE and depending on definitions ending around 1900 BCE (says Wikipedia in this case, summarising quite some time into a period I would seperate from that even earlier).
From that period we do not have that much information compared to Babylonians. Even sites like Sumerian Calendar or Living with the moon who want o capitalise on "ancient wisdom" go only so far to attribute the origin of the calendar to Sumerians, but have to resort to much later times to deliver anything concrete on the subject.
Although there is a great deal of knowledge of the mathematical systems of later Assyro-Babylonian astronomy, there is little agreement about the nature or the importance of astronomical knowledge in the earlier periods.
The central problem was addressed by Willy Hartner (quoted by Beer 1970, p. 139):
The possibility of a symbol’s original meaning falling into complete oblivion and of a new one being attributed to what would else be meaningless, can never be wholly excluded; but from the point of view of methodology, it seems objectionable to make such an assumption the starting-point of an investigation before all other attempts have proved futile.
Regarding the months:
These texts on the sun and stars are consistent with each other, but are utterly unreasonable in their lack of correspondence with reality if “day” is taken in its normal meaning, or if the months are equated with lunations. Heliacal risings, whether of fixed stars or planets, would be shifting about 5 1/4 days a year. The statements about the Sun imply precise boundaries among the zones of Anu, Enlil, and Ea, but deny the inequality of the seasons, and would rapidly cease to be true with a 360-day year. However, the Sun moves slightly less than a degree a day. It would have been entirely natural to extend the meaning of “day” to mean, also, “a degree.” Much technical vocabulary has arisen through defining a popular term more precisely, and somewhat differently. If “day” in the above statements is to be understood as “movement of the sun by 1 degree,” the statements become astronomically reasonable.
With respect to the months, there are a number of statements that imply that months were lunations beginning with the first visibility of the moon after conjunction. There are also statements of omens that imply a different calendar structure. Thompson (1900, Omen No. 249, pp. lxxvii–lxxviii) says that the moon waned on the 27th and reappeared on the 30th. More remarkably, his numbers 119–172 deal with occasions when the Sun and Moon were seen together on the 12th, 13th, 14th, 15th, and 16th of the month. These can hardly refer to a system in which the moon was full on the 14th of each month. (p213)
Our recorded knowledge of astronomical events in Mesopotamia begins with records of lunar eclipses. These eclipses are known mostly from their use (as models for predicting disasters) in the collection called Enuma Anu Enlil from the 7th century b.c. (p232)
A major problem that is raised by these letters is how the diviners anticipated the eclipse and how closely they could determine its conditions. Time, expense, and inconvenience to many people were involved. Although the possibility that the eclipse might not occur was left open, a commoner was placed on the throne, however briefly, and the eclipse did occur. It seems unlikely that substitute kings were put in every time that the diviners thought that there might possi- bly be an eclipse somewhere. From what we know both of attitudes toward the Moon and of Mesopotamian mathematical capabilities, it seems unlikely that geometric techniques were used. Some sort of cycle of local repetition seems the only likely solution. The account of Irra-imitti implies that such prediction was already occurring prior to the First Dynasty of Babylon. (p225)
- David H. Kelley & Eugene F. Milone: "Exploring Ancient Skies. A Survey of Ancient and Cultural Astronomy", Springer: New York, Dordrecht, 22011.