I can't get you there faster and more conveniently, just cheaper. TAM 100 from Monaco to the gare routiere in Nice and then walk over to the railway station and connect with TAM 23 or 99 to Nice airport. Total cost €2.60. Time to gare routiere from Monaco is 60 minutes. Time from the train station to the airport is 20-30 minutes.
You could also change to TAM 1 or 2 at the gare routiere and take it to square St. Maurice and then get the bus to the airport from there. That would add an additional €1.30 to your total fare and the time between the gare routiere and square St. Maurice is 20 mins.