To go to Mount Kinabalu National Park straight away from Kota Kinabalu International Airport would take around 3 hours. Transportation could be arranged through travel agent or ops for public transport - take airport cab to Kota KInabalu City Bus Station, then you could take Public Bus (it is van normally instead of the large Bus) to Mount Kinabalu Park. After passing Kota Kinabalu City, the ride would be through hilly area up to highland area of Kundasang. That would be could cold and expect for some hill rain in the evening.
Refer map of the journey as below or
View Mount Kinabalu National park in a larger map
or view the satellite image here
View Mount Kinabalu National park in a larger map
Route From Laban Rata to Low's Peak, the highest peak of Mount Kinabalu