Phuket old town is around 10KM from patong, but with minimal transportation options. If you didn’t want to be gouged by local taxis then this bus could be for you
It costs 40 baht flat fee. You get on then get off. No fuss.
It’s slow, bumpy and hot, but you can watch the world go by and it is by far the cheapest option. There are wooden benches, you just take a seat anywhere
To find the bus simply go to PATONG BEACH ROAD and wait for it to drive past, wave it down and get on. Anywhere along the beach road is OK. Unfortunately there is no schedule or timing for these, so you could be waiting 1 minute or 30 minutes, there is no way to know. The drop is Phuket Old Town market.