The YouniQ Hotel
What I think shall be a solution, is staying at nearby hotel within less than 30 minutes driving. Not only you can rest after hectic of long hours flight, you may recharged before taking off to your next destination especially you are anxious to be there. Besides, lack of sleep may causes lots of unnecessary discomfort during the flight. I had discomfort of headache attack every time if I fly with insomnia strikes the night before the flight.
the youniQ hotel which I dropped by during my previous flight to Melbourne was great! Comfortable interior with friendly staff at the welcome door makes you feel comfort after long flight or ride. Their rooms might not be spacious but very comfortable for over night stay.
Small but still well equipped and cozy ambiance which I really loved. They offered complimentary breakfast of toast and free flow of coffee and tea every morning starting from 5.00am to 11.00am.
Despite that, they do provide shuttle service too with a minimal charges of RM5.00 per person which I think is truly reasonable. It is about 15 - 20 minutes ride from/to the airport depending on the traffic and which airport you are arriving or departing at. Please do no hesitate to contact them via email or contact number so that they noted upon your arrival.
You can always inform them your additional airport transfer from the hotel to the airport, for next day transit upon your check in with their staff. However, there's time set for each pick up time where you can look into in their website just like the picture attached below.
Do click them here for more information. Hope you will have a great stay with them if you happened to fly or transiting from Kuala Lumpur!