As I understand it, people would not "live longer" per se. It is a misconception that, since the average age back then was some incredibly low figure, (say 35 just for argument,) that a person at 50 was a true graybeard and noone had ever heard of a 70 year old man.
The average age however, is an average, the main thing it means is that many more children died before 5 and that really old people by our standards, say late 80's to 90, were indeed much rarer than they are now, although still not unheard of. Between 5 and 70 in any era, what kills you mainly is accident, crime or war. True enough, they could be carried off by things we regard as easily curable, had intermittent plagues and the accidents, crimes, and wars were much more commonplace, but these took their toll of a population that bell curved highest at 35, (given environmental variations) just like us. The overall number was much less, however.
The effect of better hygiene would be to increase this overall number, that is, the population. You would reduce the number of infant deaths and plagues precipitately as has happened in the modern era. The most immediate effect of any population increase by one group is usually war and the expansion of this group into other areas. As they would carry their hygiene with them, and assuming they knew it to be the cause of their increased longevity, the upshot would be a population explosion, much as has happened in the present day.
The problem comes when we ask what will happen as a result. There seems, unfortunately, no good reason why farm science will necessarily expand and even if it does Malthus still wins if the population keeps growing. Ancient man was probably capable of grasping the idea that people naturally increase faster than food but even then, could he do anything about it?