Ground new wiring going into old two-wire outlet?
There is a good chance that the BX (flexible conduit) is being used as the ground back to the source or junction box. Which is fine. Just use the three wires that are shown and go to the switch. If you are going from the switch to a light, then you really do not need the ground anyway. I don't understand the loop from neutral to ground. I'd remove that and connect the ground wire. Obviously there is no GFCI in place.