I'm not 100% sure I understand the layout, but I think that your property is along the gravel path/driveway, and the wooden retaining wall is the boundary, with the neighbor owning the slightly elevated part of the lawn and the young evergreen trees? Is that right?

Yes, bamboo would have no trouble getting through that wooden retaining wall. It would follow along the base until it found a seam between two pieces of lumber or a knothole or gap of any other kind, and be through. You would definitely want to run the bamboo barrier along that wall.

Question, though, since its such a narrow space (if I'm interpreting correctly), why not just use clumping species of Fargesia? They wouldn't be running all over the place, would be easier to keep under control, wouldn't overgrow the space.

Bamboo would not need to find a seam to escape. The rhizomes would simply expand laterally and go underneath the wall and then come up. I previously saw exactly this situation where someone had a 2' high concrete block retaining wall and the bamboo just went under it and came up into the neighbor's higher yard. If you want a running bamboo, you'll simply have to have barrier on both sides. A clumper like a Fargesia would eliminate the need for barrier, but the initial expenses for your plants will be considerably higher, I suspect.

Thank you very much for the help. I will try to do a barrier, but the problem I have is there are some buried lines near the gravel, so I only have something like a few feet to do the bamboo and barrier. Do you think this will work?

Does anyone have any other suggestions for plants? I would be most appreciative for any. I liked the look of bamboo and it grows very quickly.

A couple of feet width with the length you have should be fine. In addition, bamboo are not going to damage your in ground lines. Your only concern should be not hitting them when you dig. I had the same issue when I put in barrier where my internet cable was underground: I just located the line and notched the barrier to slip down around it. In the unlikely event that the bamboo rhizomes find that little gap, it's no big deal: I'd just chop of the rhizome and the problem will have been fixed.