I have done the 6am arrival in KL a few times. Sometimes if hotels have an available room they have let me have it early, but that has usually been for a longer stay. Later this month would be getting close/into Chinese New Year. Which typically means hotels are at high occupancy so might be hard to arrange an early check in.
Close by to KLIA is Putrajaya (sort of Malaysia's version of Canberra) not much to see there except grand buildings.
Something else you could try would be a day trip to Melaka.
But anyway KL...
Shopping in Central KL
Suria KLCC - a large shopping mall in the bottom of the petronis towers. Many international brands on offer.
Pavilion - Located in Bukit Bintang, it has pretty similar shops to KLCC.
Starhill Gallery - the most exclusive shopping mall where you go to buy $300 t-shirts and $500,000 watches. There are some restaurants in the basement that are not too expensive. The toilets in the basement are an experience - they are like an underground cave
Sungwei Plaza/BB Plaza - A rabbit warren of little shops selling mainly clothes. Some quite good bargains.
Low Yat Plaza - great for computers, phones, cameras and gadgets.
Chinatown - good for fake clothes and bags, pirated dvds and roasted chestnuts
Central Market - good for tourist souvenirs, but not much else
Times Square - 13th biggest shopping mall in the world. Mainly locals shop here. Lots of good bargins
Lot 10 and fahrenheit 88 - Two smaller malls in Bukit Bintang, not much more to say about them
Food
Jalan Alor is a street in Bukit Bintang which has many roadside restaurants serving mainly local food. Most of the restaurants are Chinese, not many Malay ones. There is a place that does great chicken wings on the left hand side near at the bottom of the street. Some good seafood restaurants that do a good grilled fish. Just pick the restaurant that has the most locals eating at it.
For Malay food best to head to Kampung Baru. A Malay village right in the centre of KL. An interesting place to wander around. Or if that is too much trouble then there is a food court on the top floor of Suria KLCC that does good local food. There are also two chain restaurants - Madam Kwans and Secret Recipe that have a number of outlets - they both do local dishes as well as western food.
Skybar on the top of Traders hotel has a good view of the city. But it is popular and sometimes they restrict access to hotel guests and prebookings. There is another great rooftop bar called Lunar Bar in Bukit Nanas.
Bangsar (produced Bungsar) is a suburb about a 10-20 minute taxi ride of the central city. It use to be a bit of an expat enclave, but has diversified a bit in recent years. There is an Indian restaurant called Nirvana they are famous for Banana Leaf Curry (indian curries served on a banana leaf). There is a shopping mall called Bansar Village. There is a great winebar called WIP.
Spas
A good way to spend some time espcially if you need a bit of a rest is to go to a spa. There is a good in Starhill Gallery. Most of the hotel ones are pretty good. THere are some cheap ones on Jalan Bintang, but go with ones on the ground level (ones on the first level are typically a bit dodgy).