Put up the fence if you want to, and if the landlady complains tell her if he "doesn't listen to her anyway" then why should you have to listen to her either? If she isn't going to do anything to protect you then you'll have to protect yourself, and if you're not doing anything worse than he is doing ("not listening to her anyway") then she really can't do anything about it without taking him to task for it first.

I hope you can find a better solution than putting up a fence though. You shouldn't have to isolate yourself all around just to protect yourself from one neighbor.