The main problem with using a context-menu cursor is that it's so often left unused, people will have trouble understanding what exactly it is when it pops up.
The traditional main use of the context-menu cursor, when implemented, is to either convey that there is a dropdown menu, or prompt them to right click. (Though, again, this doesn't automatically register with users due to the fact that its so often not used.)
In the first scenario a viable alternative could be to just provide a down-facing arrow, or dropdown on hover if it is a horizontal navigation menu. In the second scenario it gets a bit more tricky, and you have to base your end solution on a few factors. Does the entire web app offer a customized right click, or is this particular area unique in that you're trying to get users to right click there? If it's the former, users should be prompted through the on-boarding process to right click and if it's the latter, a tooltip could work in prompting them.