Up Payong hill, down the other side and just across road from Tiger kingdom. Easy
We took a tuk tuk (family), driver waited and then took us back. We just told the driver 'Go karts, near Tiger kingdom' (it's literally next door). You pass a dodgier one on the way... you don't want it.
Get quick answers from Patong Go-Kart Speedway and Phuket Offroad Fun Park staff and past visitors.