Mt Kota Kinabalu is within a 2hr drive of the city itself. The climb is only just over 8km long, but it is a steady and constant uphill and extremely tiring! Within two days, you will go from sea level, up to 4,000m and back down again.
The first day invloves getting there and walking the first 6km (takes between 4-6hrs). You stay at Laban Rata hut and get a few hours kip before starting the summit at 2am to get up in time for sunrise. On your way down, you can opt to do the Via Ferrata - the world's highest: http://www.mountaintorq.com/
Climbing Mt KK is quite expensive and you have to book way in advance as there is limited accommodation and no camping is allowed, but if you go to the region, this is a must do activity!
For those rock climbers out there, the North Face did an expedition to Mt KK in April 2009 - EPIC. Check out this link for videos and how it went: