When I did this with my family a few years ago, we flew into Darwin and stayed a night at a hotel near the airport. The next day we picked up a hire car and drive to Edit Falls. We left our car at the carpark and let the people at the cafe know our rego details and when we planned to be back at the car. We then met a shuttle Bus that drove us to the trail head at Nitmiluk (Katherine) Gorge. The drive and shuttle took up most of the day so we stayed in a cabin for the night at the trail head. Next day, started the Jatbula.
After 5 days hiking, we reached the end of the hike at Edith Falls and were thrilled to have our hire car sitting there. It was so nice to get in the car and leave when we wanted to. It was also nice to have a swim at the falls and change into a fresh set of clothes before departing. I know some people catch a bus all the way from Darwin but we liked the flexibility of having our own car.