Look for the black & white question marks on the map or minimap. Approach the person at this location to initiate the cutscene (you may have to hit (B) button to initiate).
After the person gives you instructions you will have a purple circle on your map. Head to that location to find the next part of the mission. You can view your journal from the main menu to review the objectives as well.
Once you have completed the mission(s) go back to the original stranger location (now marked by a purple question mark on the maps), approach them to initiate the cutscene. You will get the achievement once the cutscene ends.
I completed the 'Water and Honesty' mission with is initiated by a man south of Armadillo. It only had one mission objective; obtain a deed to some land to the north. Travel to the mission objective (the purple circle on the map), kill or pay the man for the deed, and then return to the stranger spot south of Armadillo (purple question mark). I completed this in less than 5 minutes. I happen to have saved my game near the stranger spot the day before though (random, not planned).