If we are talking of lower Paleolithic hominids did they not eat their
marrow bones etc  by the side of waterholes or streams? In which case
they could drink by lapping or cupping their hands. Cupping their hands
would allow them to keep an eye on approaching predators so would be the
better bet.

If we are talking about  Upper Paleolithic people they could make
containers of leather, since they had awls and needles, or of birch
bark. Leather bottles were in use until the eighteenth or nineteeth
century in Europe and birch bark  was a very useful material  for
containers which could be made waterproof with resin or "tar" in
Northern Eurasia and North America into this century.
It would be interesting to find out if rock shelters caves and open air
sites were situated  close to sources of water. Though in the case of
springs and streams  these would have dried up since.

M E Wood
Christchurch New Zealand