Trains run directly from Nice Ville to Monaco-Monte-Carlo, with journey times of between 13 and 23 minutes. On Sunday, 27 May, the trains leaving between 0800 and 1100 depart at 0812, 0820, 0836, 0910, 0944, 1005 and 1051.
Those times are taken from the German Railways site (which has rail times for the whole of Europe):
http://reiseauskunft.bahn.de/bin/query.exe/en
The French Railways site doesn't yet have the services for the end of May available but, by looking at the services for an earlier date, it can be seen that the single fare is between �2 and �3 (depending upon which train you travel on):
http://www.sncf.co.uk
Official F1 website information for Monaco:
http://www.formula1.com/race/destination_guide /774.html
Official Nice tourism website:
http://www.nicetourism.com/GB/som.html
Chris