For morning I would suggest getting a cab although Haryana roadways bus service (very good frequency) is there on this route from Gurgaon Bus stand but not from Metro station. Buses going to Bhiwani or Jhajjar will make pass from sultanpur but there is a catch to it.
You'll have to specifically ask if the bus will go via farukh nagar or not coz these days 50% buses are taking off route via another route (towards Badli - my place) giving Sultanpur route a miss.
Regarding your query about shared auto, these will mostly take you only upto Basai Wetlands which is another popular Bird site/ watching area on midway to sultanpur. Sultanpur sanctuary is like 10 kms from that place.
Self conveyance is best for such trips though as you could cover both sites.