Which minibus goes from Kaspiysk to the Sarykum dune??
Khasavyurtovskaya, as an option, can be taken up to the turn, but it is expensive, rarely runs, and it's better to reserve a spot in it in advance by calling the driver.
It’s definitely not possible to drive straight to the dune. There is an option to get to the federal highway, from there it’s quite a walk. I advise leaving the idea of going to the dune by minibus.