What would be the best way to get to the Nice airport from
Antibes early in the morning. We need to be there by 6:00 am and the taxi prices are a fright
I don't actually know about transportation from Antibes to Nice Airport, but I think you incase you couldn't find any, checking out the option of going around midnight (or check when is the last bus) to the airport after a night out in Nice you can probably find a place to crash while waiting even a chair. Not only you'll be there in time in the morning for your flight, but also you'll save money for a bed.
If your departure is on a Friday, Saturday, or Sunday morning, you could take the last Noctam'bus from Antibes to the airport. Leave Antibes at 03:00 and arrive at the airport at 03:46. (Information from the Nice airport site.)
If you're open to sleeping in the airport, go here: www.sleepinginairports.net/europe/nice.htm.