Some of the Children's Tylenol is essentially mixed in with sugar (e.g. the cherry, bubble gum, fruit, etc flavors) so Tylenol should be OK with your sugar syrup.
And a child that starts with vomiting first, then diarrhea soon afterwards does seem to point toward a viral gastroenteritis as one of the strong possibilities.
But appendicitis classically comes with specific abdominal pain (for example, pain that starts near the belly button, then moves down toward the right, and now localizes to the "right lower quadrant" of the abdomen). Other classic symptoms are fever, vomiting, and anorexia (no hunger/appetite).
But in children, these classic symptoms may not be present, though if your child has pain that is worst when pressing on the right lower part of the abdomen, and if he has "guarding" (the involuntary tightening of the abdominal muscles when you press), and rebound tenderness (the pain is also bad when you let go suddenly after pressing in the area), then it would be wise to have this evaluated by a doctor in person.
Of course, keeping him well hydrated in any case of vomiting and diarrhea is important to prevent dehydration.
The diarrhea itself may be a key symptom that points toward a viral gastroenteritis, rather than classic appendicitis, but the only real way to know is to have a doctor examine him in person.