Which one of the following drugs increases gastrointestinal motility?
Neostigmine belongs to group Anticholinesterases (anti-ChEs) which are the agents which inhibit  ChE,  protect ACh from hydrolysis-produce cholinergic effects in vivo and potentiate ACh, both in vivo and in vitro. Quaternary ammonium  anti-ChEs like neostigmine have additional direct action on nicotinic cholinoceptors. 
ACTIONS (of Ach)
Smooth muscle in most organs is contracted (mainly  through  M3 receptors). Tone and peristalsis in the gastrointestinal tract is increased and sphincters relax leading to abdominal cramps and evacuation of bowel.
Thus, Neostigmine increases gastrointestinal motility.
Reference: Essentials of Medical Pharmacology Eighth Edition KD  TRIPATHI  page no 114,117
Glycopyrrolate
Fentanyl
Atropine
Neostigmine