Short answer:

There's a lot of interest in geography in the ancient world, and plenty of description of rivers surviving in our oldest ancient sources. Wells and irrigation technologies in many places around the world in ancient, and indeed in prehistoric times.

Discussion:

Civilization begins with agriculture, and we find complex irrigation systems, dams and aqueducts in antiquity-- the Romans are particularly well known and skilled at this, but in fact the interest is widespread.

Do "their insights agree with modern hydrology?" -- you'd have to be a little more specific. They didn't run "perc tests" or the like, if that's what you mean. But a Roman purchasing rural land would look into the availability of water from natural springs, either on his own property or purchased from a neighbor-- we have records of these legal agreements.

There's a great deal of interest in natural springs, and the Romans in particular characterize them-- they're interested in the different properties (and indeed temperatures) of different water sources.

On a theoretical level, they had models which were both in some senses consistent with modern hydrology, and others which are inconsistent. So you'll find some suggesting that the source the water in rivers is rainfall, which is correct, roughly. There's also another model, also dating back to the Greeks, of water being "purified" underground. . . they imagined water being pulled in underground channels from the sea and purified. That's obviously mostly incorrect.

Most impressively, you can find several Greek writers who have a conception of the water cycle-- Thales of Miletus, Xenophanes of Colophon and Anaxagoras all have something like this in their work.

When did ancients give rivers their names? Long before historical records begin, in most cases. The Tiber river in Italy, for example -- that name seems to predate the Romans and any written record; there's a story of a King Tiberinus, who was said to have drowned in a river then called the Albura . . . henceforth it was the Tiber, or so we're told. Tiberinus isn't historically attested, so we're in the realm of myth and pre-history here, but it was something the Romans said, in any event.

Ancient civilizations paid careful attention to river floods-- that's particularly true of the Nile, for example. The flooding of the Nile was both seasonally predictable and a great boon to soil productivity . . . the ancient Egyptians knew it was coming and anticipated it. We have their records of flood heights recorded as early as 2000 years BCE (eg four thousand years ago).

The world's oldest dam still in operation was (may have been destroyed in the current conflict) at Lake Homs in Syria-- this dates to 1300 BCE. We find evidence of dams thousands of years older that no longer are functional -- the Jawa Dam in Jordan goes back to perhaps 4000 BCE

There is a vast literature. Dams and irrigations systems leave some of the most durable archaeological evidence-- often from peoples for whom we have very little else.

Sources:

Jacobson, Howard. “Artapanus and the Flooding of the Nile.” The Classical Quarterly, vol. 56, no. 2, 2006, pp. 602–603.Bell, Barbara. “The Oldest Records of the Nile Floods.” The Geographical Journal, vol. 136, no. 4, 1970, pp. 569–573.

Kay, Shirley. “SOME ANCIENT DAMS OF THE HEJAZ.” Proceedings of the Seminar for Arabian Studies, vol. 8, 1978, pp. 68–80.

Crown, Patricia L. “Water Storage in the Prehistoric Southwest.” Kiva, vol. 52, no. 3, 1987, pp. 209–228.

Harrower, Michael J. “Hydrology, Ideology, and the Origins of Irrigation in Ancient Southwest Arabia.” Current Anthropology, vol. 49, no. 3, 2008, pp. 497–510.

Liu, Bin, et al. “Earliest Hydraulic Enterprise in China, 5,100 Years Ago.” Proceedings of the National Academy of Sciences of the United States of America, vol. 114, no. 52, 2017, pp. 13637–13642.

CHRISTIE, JAN WISSEMAN. “Water and Rice in Early Java and Bali.” A World of Water: Rain, Rivers and Seas in Southeast Asian Histories, edited by PETER BOOMGAARD, Brill, 2007, pp. 235–258.

Rawlins, Dennis. “The Eratosthenes-Strabo Nile Map. Is It the Earliest Surviving Instance of Spherical Cartography? Did It Supply the 5000 Stades Arc for Eratosthenes' Experiment?” Archive for History of Exact Sciences, vol. 26, no. 3, 1982, pp. 211–219.

Making the Circle Round: Perceptions of Hydrology through Time.” in Following the Water: Environmental History and the Hydrological Cycle in Colonial Gippsland, Australia, 1838–1900, by KYLIE CARMAN-BROWN, ANU Press, Australia, 2019, pp. 21–38.

R Nace, "General evolution of the concept of the hydrological cycle" in Three centuries of scientific hydrology: Key papers submitted on the occasion of the celebration of the Tercentenary of Scientific Hydrology, 9- 12 September 1974, UNESCO, Paris, 1974