Urban Romans, we're told, took a bath every day. They did not use soap. Instead, they oiled themselves and scraped off the oil, along with the dirt, with strigils. What did they do about their hair, though? You can't use a strigil on your hair. Simply rinsing it in water would have resulted in plenty of dandruff and not much else: the hair would still be dirty and hanging in greasy strands. What did they use to get it clean?
They used lye soap which is made by combining ashes with lard or other oils and fats. This kind of soap was known from ancient Egyptian times. It was customary in Rome to always wash your hair on August 13th in honor of Diana, but they washed it other times as well, obviously. The Romans bathed a lot and they (especially the women) would wear little caps to prevent any unwanted water or oil from getting into their hair.
Caesar commented on the barbarians he fought in Gaul that they washed their hair with lime water which made it very coarse.
I'm currently doing personal research on the history of hair hygiene and from what I've seen, throughout European history the way it works is :
daily (sometimes more) brushing and combing of the hair with clean brushes and combs (the finer the better) it is the base of hair cleaning in all of history before the" big chop" of the twenties
protection of the hair (although uncovered hair seems to be more popular that in the middle ages and I have no idea what a they slept in)
occasionally (and often not advised and the thing that changes the most) degreasing of the hair which could be done dry (with barn or starch) or wet (with soaps, saponins, tanins, eggs, alcohols, acidity ,etc)
Soaps, saponins, lye, clay and eggs were available to them but I have not find any sources I trust mention their use for haircare in rome specifically so if you were to be transported there it would be some options you have but if it wasn't for that mention that hair was washed on Aug.13th I would have assumed they just combed and covered the hair
Also, from my personal experience, having your hair up all day long really helps in keeping it less oily (plus I would assume the wool used to style the hair would have soaked up some of it)
Edit: I feel like it could have been part of Roman mentality to mock a culture for doing something and then copying them, they were proud of the "cosmopolitan-ness" of their culture but mocked anyone who didn't conform to the Roman way of doing things