The most efficient way to visit Gunung Mulu is to fly in for an over night stay from Kota Kinabalu. They have daily flights into Mulu via Miri and the plane lands at about 3.30pm. You can do the Deer Cave right away and watch the bats come out in the evening. It will take an hour walking through the forest on raised plank walks. The Clearwater cave can be visited the next morning via a 30min boat ride on the Melinau River. This is really pleasant and peaceful ride.
After lunch, there is a flight back to Kota Kinabalu at about 2pm. Well worth the effort but bring rain coats.