I would not use grease. Swarf will stick to it and make a splendid grinding paste.
Not much oil is needed on change gears — the load on them is minimal and they are cast iron so "self lubricating" to a certain extent.
Any old oil will do ok on the back gear. Whatever you have in your oil can. That said, I use spray can motorbike chain lube because it sticks to the gears but is not sticky (once dried) and makes minimum mess.
Be sure to oil the pulley where it spins on the mandrel when in back gear. Not sure on your model if/where the oil hole or fitting is. On some it is in the side of the pulley at the large end.