I actually just got back from doing something similar to this today. We weren’t at a hotel but flying in/out of TF Green. There is a RIPTA shuttle from the airport (and possibly other locations) to the ferry terminal and then a ferry from Providence to Newport. The ferry was actually a nice scenic ride in the bay and was $30 for a family of four. I can’t remeber exactly how the price broke down for adults vs kids.
But, the ferry ride is out of Providence and you'll be backtracking to get there.
I'd take a cab, uber or lift and enjoy the extra time in newport