About 20 Italian Cyprus, or 3 Ficus Benjamina. they will need water, so you need to figure that out. They will grow tall and block the neighbor's view.

A secondary fence consisting of about 3 to 4 panels could be constructed to block the view and it would create a backdrop for a courtyard. You could plant on the side by the seating area at ground level, and on the opposite side facing the neighbors, plant taller plantings like chaste tree or crape myrtle with under-plantings. The wall you have created would look great with a path between the house and the wall, then you would naturally walk around the wall to check on the plantings on the backside of the wall, and that area could become the strolling garden area that reconnects to the patio on the opposite side, over by where a barbeque might be placed. Definately an interior walled courtyard would be nice.

If you can't see them, they can't see you ... all you need is a few big bushes or small trees to the right of the tree.

Also, a pergola, arbor or other garden structure in that spot could probably block that house's view of your yard. Something with a trellis and vines could immediately give you a sense of privacy without the root problem.

Where do you live? That would affect the choices of plant material.

I agree with lazygardens, a pergola or arbor would be very nice and make a useful outdoor living area! The yard looks a little empty, at least in the photo, and a structure of some type would cozy it up a bit.
You could use trellis or outdoor blinds to block the view at least until your plants are large enough to do it. Or just use the trellis/ blinds and no worry about roots (although some plants would be nice).
Let us know what you decide to do!

I do not know what the neighbor is like, but Maybe one could talk to the neighbor and ask him if he would consider planting a bush or tree on the uphill side of the fence. I imagine that he might want some privacy too. Then there is always gorilla gardening...LOL