The current plan of the neighbor apparently includes a hedge but no fence. Rainwater infiltrates on his property. How is it with a garden shed? Does it have to be drained, or can he just let it run off (on his property)?
Garden house, gutter, downpipe, rain barrel, and you already have watering water for indoor plants, tomatoes, and co. His garden house should also remain as boundary construction with the roof overhang on his side, because you want to build your garden house as boundary construction with roof overhang on the same spot towards him.