Shimano Tourney "Rapid Rise" Rear Derailleur Installation
AS you have found out Rapid Rise systems reverse the derailleur spring and direction of positive pull from the shifter. The idea was to make it easier for inexperienced riders to quickly change to lower gears.
You just need to reverse what you would do on a normal derailleur setup:
If you detach the cable the derailleur cage should want to move to the most inboard position over the largest sprocket as you turn the cranks. If the cage does not do this then you'll need to check for an obstruction or the lower derailleur limit adjustment may be incorrect.
Put the shifter in the lowest gear position, then attach the cable. If you are unsure about which end of the shifter range is low or high - when the shifter shifts towards lower gears it should pay out cable, when shifting to higher gears it should pull in cable.