You only need several cards if you're using unlimited cards. If you have a weekly pass or fun pass, it can only be swiped once every 15 minutes or so.
If you purchase a $10.00, you can swipe as often as you like until the card reaches zero.
You can take the airtrain from JFK to Howard Beach then transfer to the A or C to 42nd Street. (Times Square) The A is express and the C is local.
When is your trip? Transit is doing major track work on a portion of the A and C lines for a few weekends.
If you take the train during that time, you'd have to get off at Utica Ave. Once you walk upstairs, there will be shuttle buses to take you downtown to Jay Street. You get back on the train at Jay Street.
There will be three buses there. The B25 ( don't get on that bus). A local shuttle. You can take that one but it will take forever to get downtown. There will also be an express shuttle. That's the one you want. Once that bus leaves Utica Ave, it won't stop again until it gets to Jay Street.
There are plenty of transit workers at Utica Ave to answer questions and direct you to the correct bus.
Now if it were me, I wouldn't deal with all of that. I'd just take a cab from JFK.
If you want, I can look up the repair dates. Maybe your trip falls after the work is completed.
