Care and maintenance of wood fence that is rotting because of damp leaves on the other side
Wood and water don't mix. If you want to make wood last, you have to keep it dry. So no good answers unless you can convince your neighbor to remove the rotting debris. Assuming you and the neighbor share the cost of maintenance and replacement  of the fence would entice them to take care of this.
Regarding preventative treatments, there are some do's and dont's:  Don't use a solid surface covering like Varathane or Ureathane:  Outdoors that finish will crack, chip, peel and look terrible. Paint might be an option.  My preference would be a penetrating oil with additives to prevent mold, mildew and rot. Whether paint or oil:  You'd have to use a wood brightener (it's like a bleach) on it first, then pressure wash, wait until very dry and apply the oil.   The tough thing is for this to be effective, it needs to be to BOTH SIDES of the fence, which may be at least impractical or impossible in your case.
Depending upon where you are, bare wood outside isn't the best option...here in western Washington State, I would never install a wood deck or fence again. It just doesn't last.