Sounds like the fence is on the neighbor's property. If the patio goes right to the edge of the property line, everything afterwards must be off it.
Anything you build needs to be 100% on your side of the property line. GP_Matt has it bang on though - you'd need a survey to be 100% sure of where the line is. My guess is $250-$500 range (2 guys, 1-2 hours).
As for the fence itself, here's hoping you have cool neighbors. It'll make life so much easier for you!
__________________
|