I've never been asked to produce my passport there, but theoretically since it is a proper border post they could ask you at any time, so its better to take one with. But they are used to hikers and it's not like the border is right there, it's further away. There is no Lesotho counterpart border post where you are going either.
Not sure that there would be much water directly at the summit of the pass at this time of year, but I don't think you'll struggle to find water a short distance further in, from the Thamathu River flowing into Lesotho.