Neem oil is to kill insects - it works by bridging across the spiracles through which they obtain oxygen . No good for anything else ( except bunging up your sprayer when it dries out and turns to wax ).
But where is the fungus ? I can't see any ! Those slightly darker patches on the older leaves, not the new ones - could be any of several things, but a source of worry is not in that list.
( for Sheryl and other well-wishers - my wife is expected to be out of hospital in a few days now... after 5 weeks. Hallelujah ! )