The are 4 different ways across 2 unique modes of transportation to help you travel from BBY to Boston. The fastest way out of Back Bay towards Government Center is by public transit, as it is the quickest to the Boston area. This doesn't necessarily take into consideration traffic or weather - so these may impact some of your decision making. It's common to find tickets for as low as $2. MBTA operates from BBY to Boston, so you book your trip with them! Take public transit from BBY to Boston, it's your best chance!
The earlier you can book your trip to Boston, the better. There is usually no need to book these sort of trips from BBY to Boston in advance. Rideshares should be available on demand all hours of the day. Public transportation prices do not fluctuate like flights. Instead, they're either flat rate or priced by distance traveled. If possible, you can book your tickets via an app or alternatively at the station upon arrival.