there's probably not a shuttle from the amtrak station but instead you can walk five minutes away to Kennedy Plaza and get a bus to the airport. check www.ripta.com for details.
or
if you are leaving Thursday night i can probably give you a ride as the airport is not too far from my house anyway.
i usually drive to Route 128 which is a closer T stop that has more trains. it costs $3.50 to take the train so if you want to take the train with me to route 128 it will take approximately 20 mins to get there from back bay and the approximately 45-60 mins to get to the airport from there and that is figuring in slight traffic once we get to the downtown providence area of 95 which jams up like crazy between 5-6.