As far as I understand from you question and images, I think Revolved Boss/Surface is the easiest way to get it done. Other options I can think of Lofted Boss/Surface or Swept Boss/Surface. It depends on what you want to do with it afterwards.
When you are extruding, there is an option to apply a directional vector for the extrude. Set it to match the taper of your cone, and you should have what you are looking for.