Is a good PhD student or researcher expected to spend some part of their day/week regularly reading new research papers/pre-prints?
There are pages like http://www.arxiv-sanity.com/ to filter the mass of papers on arXiv and give you the chance to read what interests you most. Furthermore, you can also go through all recent papers on your phone (instead of browsing instagram, reddit or whatever), read the abstract and mark everything that might be interesting to later read. I'm not sure if it is expected, that might strongly depend on your institute, group or advisor, but it is surely possible to do it in nicer ways than by simply clicking through arXiv directly.

I know good researchers who check arXiv daily, I know other just as good ones that haven't looked at it in months, so there also is no general rule there for after the PhD it seems.

Btw, if anyone knows of other nice ways to filter the arXiv feed, feel free to share/add, I'm not yet 100% happy with the ones mentioned that I currently know.