You don't need any specific ramp pass for Captree, or probably any of the state parks. You can just pay everytime you go, otherwise, you can get the Empire Pass for $59 and use all the state parks with unlimited access. I picked up the Empire Pass at the Robert Moses Park Police office.
You can get to Captree by taking the Parkways. You can only trailer technically from Merrick Road/Montauk Highway South on the parkways, but I've never had a problem, and have even seen cops, while trailering off of Sunrise. Just be wary of the bridges for Loop Parkway and Merrick Road on the Meadowbrook.