I did this very trip several years ago. Door to door was 55 minutes, since I got lucky and made quick connections. Max time = 75 minutes.
The cost for the bus/subway fare is $2.25 + $5 for the Airtrain.
Why did I do this? Glad you asked. UA at LGA was looking for bumpees. I volunteered and suggested that I be re-routed JFK/SFO. Of course they looked at me like I was nuts.

Do any of you think this was nuts?
They give me my $250 bump voucher. I ask for cab fare LGA/JFK. They say no. I say what about a $50 UA voucher? Well, it seems that this request was easier for them to process.

As long as I was asking, I asked for an upgrade on my JFK/SFO flight. The agent told me she'd deposit a CR1 in to my account to cover the upgrade. There was just one problem, she deposited a SWU instead.
I know this is off-topic, but since this is Milepoint, I know that you appreciate the store.
That's the story of my LGA/JFK transit. It was profitable.
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