I will be arriving JFK mid-afternoon before rush hour, and need to get to Armonk without having to rent a car. No luggage, just carry-on.

Should I take the airport bus to Grand Central and connect on Metro North? (What would be the closest station to Armonk?) Or should I just try to book the JFK-Westchester shuttle service? (I recall taking the van from Prime something--are they still in business?) What would be a better choice, in terms of costs and time?

The nearest Metro North stop is North White Plains. You will need a taxi from there.

If you're going to an IBM function, you might check with them for a recommendation.

The cheapest way to go is JFK/Airtrain/Subway/Metro North, but it might take you 3 hours depending on how you hit the schedules.

The fastest way is by limo service, but quite a bit more expensive.

Another option is a shared ride van service like Go - I think that's the current name for the Primetime service to which you refer.

http://www.2theairport.com/shuttles/jfk.html

If you don't care about the time, multiple train connections, then taxi you can use the route gb describes - lots of people do.

If your destination is close to NWP or WP Metro North then that's not a terrible option.

If you're going to a hotel or business in Armonk I'd prefer to get dropped off right at the door straight from JFK, although you may have other passengers/other stops along the way with a shared ride van.

Maybe it depends on where in Armonk, but when I go to maps.google.com and map Armonk NY and then look for nearby train stations, Chappaqua is much closer than North White Plains.

Here's a car service
http://www.airportspickup.com/we_rates.html