According to the Apt tourism site, there's no direct service from Nice:
http://www.ot-apt.fr/us/ot/ot.htm
Your best route is probably to take the train to Marseille and pick up the bus from there. The train service is rather irregular. For example, there are weekday morning trains at 0718, 0833 and 1008 but the next train doesn't leave until 1330. The journey time is around two and a half hours and fares are quoted at either �20 or �22.50:
http://www.sncf.co.uk/
Unfortunately, it looks as if there are only two buses per day between Marseille and Apt. The first service, at 0700, is too early for a connection from Nice. That just leaves the 1700 service which seems to require a change of vehicle at Pertuis. The arrival time in Apt is 1915. (The 1330 train from Nice arrives at Marseille at 1602, so that might be your best connecting service).
Theoretically, you can travel on the daily bus service from Digne. This leaves at 1130 and arrives in Apt at 1325:
http://www.ot-apt.fr/ot/car.pdf/Horaires_Apt_D igne.pdf
However, it seems that the only service from Nice to Digne doesn't get there until 1130, so it's a very risky connection!
http://otdigne.cyber3c.com/index.php?option=co ntent&task=view&id=71&Itemid=108
Chris