Mike,
Your best bet on that route is probably a departure from Monterosso at 06:54, arriving Nice Ville at 11:51 (time 4H:57M, one change at Ventimiglia, reservations compulsory on both legs). That's about the shortest trip, and the one I'd suggest. While that means an early start for the day, it will get you to Nice at a decent hour.
You might ask your hotel in Monterosso if they can fix a "bag breakfast" for you, as you won't be able to get breakfast at the hotel and there are no facilities at the station in Monterosso that will be open that early (or you could pick up a Panini or something the day prior to travel and have the hotel store it in their fridge until you leave). I've done that on many occasions.