To unlock the staff ghosts, you must beat these times (special cup is unknown):

Mushroom Cup
Luigi Circuit - 1'29"000
Peach Beach - 1'23"000
Baby Park - 1'14"000
Dry Dry Desert - 1'53"000

Flower Cup
Mushroom Bridge - 1'34"000
Mario Circuit - 1'44"000
Daisy Cruiser - 1'55"000
Waluigi Stadium - 2'02"000

Star Cup
Sherbet Land - 1'26"000
Mushroom City - 1'53"000
Yoshi Circuit - 2'02"000
DK Mountain - 2'15"000