Is there a shuttle from the hotel or close to the hotel that brings me to the Niagara Falls area (the falls itself, around Rainbow bridge maybe)?

You can check with the NFTA this is our bus transportation. You can take an Uber, Lyft or cab there. We do not have a shuttle from the airport to go the Niagara Falls. You may want to check with the Buffalo Niagara Airport when you arrive.

Is the airport shuttle free or what is the fee?

The hotel shuttle just runs to and from the airport. I don't know of any other destinations.