The cheapest way is to walk up Seventh Avenue to 32nd Street, and then walk one block east to Sixth Avenue (Greeley Square). Enter the subway, buy a Metrocard for $4.50, and board a Queens-bound F train. Get off at Kew Gardens/Union Turnpike (about a 30-minute ride), and then board a JFK-bound Q10 Limited bus, which will get you to JFK Terminal 4 in about 35 minutes. You can then ride the "All Terminals" AirTrain free of charge for the five-minute ride to Terminal 1. You will get a free transfer from the F train to the Q10 Limited, so you will have $2.25 left on the Metrocard for future use.
A faster, but more expensive, route would be to walk up to 31st Street, enter Penn Station, and take a 20-minute ride on the Long Island Rail Road ($6.25) to Jamaica. At Jamaica, board the AirTrain ($5.00) for the 10-minute ride to Terminal 1.
As long as your bus from Boston is more or less on time, the cheaper subway-bus route should get you to Terminal 1 by about 1:00 PM.
The cheapest way is to walk