Your second pic looks like you're trying to build two enclosures.
If you and the neighbor get along and he's willing to help, just take enough of the fence apart to allow you to dig up the rotten post and concrete, then replace with a new post. THAT is the best way to do this...
HOWEVER, if you have rotten fenceposts and you have neighbor problems as many of us do, you COULD dig in new fenceposts in the center of your existing panels and live with the look of the extra 4x4. Not the BEST idea, but it fixes the stability of the fence without having to battle with a bad neighbor...and you can easily do it yourself.
Just dig a hole 12 inches in diameter and about 2 feet deep in the center of the panel you want to strenghten. Cut the 4x4 to length , stick it into the hole, plumb it in both directions (see the link I posted earlier) brace it plumb, then mix and pour one 60 lb. bag of concrete post mix into the hole. Be sure to use a piece of 2x4 or something similar to "rod" the concrete around the fence post to help remove any airpockets. Then nail the top of the fence to the 4x4.
Normally, I'd cut the bottom board (horizontal board) and insert post there, but yours looks like 1x4...difficult to toenail into. You MAY have to live with the 4x4 sticking out on your side. To me, that's an unusual fence...built with what looks like 1x4 stringers instead of 2x4 stringers...but hey...you ARE in Canuckistan...