Any hotel the weekend of the marathon is VERY expensive. So if you are on a budget, you are probably not staying within a mile of the finish. Just remember to factor that in. $500/night minimum kind of rates.
This would be my recommendation for you -
Stay in the Seaport. Look up the hotels in the Seaport area. (Westin, Renaissance, etc..) They will be expensive .... no more than Copley Square area, but they COULD be less.
But when you stay there, you can simply hop in a cab to get there after the race. It should be about a 15-20 minute cab ride. You'll be at the hotel, and you can shower/change/pack. Then it's about a 10-15 minute ride from there to the airport.
If you are running as fast as you plan, then the finish area will not be a complete mob scene when you finish and it should be a lot easier to get away.