How do I join my horizontal slat fence with my neighbor's fence (vertical fence)
Before you start, talk to your neighbor first, get a feel for any concerns if any they have. Then inspect the post to see if it would hold a section of fence. Then go to it. 

EDIT: Ok lets try a different approach, dose your property line split the post, (red line), or meets it (green line)?

 

This will make a difference. I would also consider placing a stationary panel first because the swing of you gate would be hitting the curb on the driveway.