There are tuk-tuks running most everywhere on the island, but Jungle Beach Resort is located some way back on Nai Harn beach, and I'd doubt many tuk-tuks would go there.
The remoteness of the resort is one of the reasons people like it, but if you want to get around the island I'd suggest hiring a car for a few days. Buses to Nai Harn stop at the Phuket Yacht Club, and from there it's still at least one kilometer to the resort - and that's over a steep hilly road.
There are no normal prices for tuk-tuks. Drivers try to get as much as possible so it's best to bargain. Expect to pay 700-1000 baht a day for a Suzuki Jeep, or a bit more for a sedan car.
Hope this helps.