Spraying an apple tree

I'm sure everyone has seen me moan about the neighbour's apple tree at my back fence line. It has been dropping diseased or rotten apples on my lawn for about 3 weeks now. They don't prune, spray, or do anything to maintain this tree.

Is it possible to get healthy apples from one half of a tree?

Sounds silly I know but I'm thinking of the "if you can't beat em join em", way of doing things.

Perhaps if I sprayed whatever is accessible from my side, at least I might get some healthy apples from the tree instead of useless spoiled fruit?

Or would it be useless to only spray part of the tree?

Also, I've read that spinosad should be used or dormant oils or other chemicals (carbaryl) are what should be used. Lots of different opinions. Does anyone have first hand experience on what works?

many thanks
Lori