In comparing the two maps, it doesn't look like there is much of an easy way to get from Coney Island to LIRR. In Brooklyn, the LIRR stops at Flatbush, Norstrand, and East New York.

From what I can tell the closest common connection between the two is Norstrand. But you're looking at probably two trains and a shuttle transfer.

You could probably take the Q to Prospect Park, then take the shuttle from there to Franklin Ave. One stop on the C train from Franklin Ave is Norstrand where you can pick up the LIRR.

Perhaps someone knows an easier route? I just came up with this from comparing the two maps.