- JFK AirTrain to Jamaica Station ($5, ~15 min)
- Inbound Long Island Rail Road service to Atlantic Terminal ($6.25, ~20 min)
- JFK AirTrain to Howard Beach Station ($5, ~15 min)
- A Train (NYC Subway) to Hoyt-Schermerhorn Station ($2.25, ~30 min)
Well, that's not entirely true. The truly cheapest options are to take the B15 bus and transfer to another bus somewhere in east Brooklyn, or to take the Q10 bus to Lefferts Blvd-Ozone Park Station and take the A Train from there to Hoy-Schermerhorn. Either ride would be $2.25 for the whole trip with a MetroCard. These are a much, much longer routes (up to two hours) through some very, very boring areas, and I would not advise a visitor, who has limited time, to bother with them.
If you want to play with some options yourself, try the MTA Trip Planner.
I'm not familiar with the MTA pass options, but you can save money by buying multi-trip MetroCards. MTA has a handy guide to MetroCard options based on the number of rides.