I feel like private charter prices in the Virgin Islands have gone up considerably over the past several years. We've used Palm Tree Charters a couple of times and really like them. They were very reasonably priced compared to others when we shopped it, but it was still expensive. Our charter last year was $1,250 plus fuel and tip for up to 6. If you just want JVD and no other stops, there is also a ferry that goes from Cruz Bay to JVD. There are "taxis" on JVI that will get you from Foxy's to Soggy Dollar which I think are the best bars on the island. We've not done the ferry so I can't comment on how it is.
https://www.interislandboatservices.com/ferry-trips/red-hook-cruz-bay-jost-van-dyke/Lime Out is awesome. Great tacos and craft cocktails. Salty Daze runs a shuttle to Lime Out. The shuttle does sellout so you definitely need to reserve ahead of time. Also, Lime Out gets really crowded so I'd recommend the the first shuttle to ensure you get one of the floating couches.
https://saltydazeusvi.com/limeout/