Arriving in 5am, I think that you should take your time by taking the Skybus to KL sentral which will cost you 11 ringgit (2-3 dollars). Or if you're racing against time to have a quality time in KL, you should take the KLIA Express. A convenient and quick transportation to the city with a warranty of 31 minutes from the airport to KL Sentral but would cost you a hefty 55 ringgit(13-14 dollars) to KL Sentral, or 38.40 ringgit (9-10 dollars) to Bandar Tasik Selatan. Whichever way will require you to transfer to a KL Komuter by taking the Batu Caves-Klang line to Batu Caves (2.40 ringgit or 0.5-1 dollar from Bandar Tasik Selatan or 2 ringgit or 0.5 dollar from KL Sentral). But just to remind you that by my experience traveling to KL, I've been on a commuter at rush hours, the coach will be full of people with long commutes alighting in KL Sentral and other stations downtown, so expect yourself to be in the middle of commuters, but then again, it depends on the day of your visit of course, it should not be that packed on weekends.
You should spend around 1-2 hours in Batu Caves, depends on whether or not you will climb the stairs all the way to the cave or not, you should be done and back downtown (KL Sentral) by noon time. There are plenty of things to do like experience the hustle bustle of the KLCC (it's where the twin tower is at), or go to Bukit Bintang for a short shopping experience. For cheap food, you could go to Lot 10 Hutong in Bukit Bintang, it's where the street food is collected and placed in one comfortable food court. Well a day might not be enough, but then again, it depends on what you want to do and see in KL. Then the recommendations are varied.