We did it on the cheap and here's how we did it.
The taxi to Old Phuket Town
From landing at Phuket international we got a taxi right out front. We opted for a pay before you go taxi service, they have a desk outside the main entrance. We've heard so horror stories about people getting ripped off with metered taxis that never switch on their meters, people being abducted for human trafficking, etc. So do take care when choosing a taxi. The taxi cost us THB650 / ZAR260 or $20. Considering that it was a 40-minute drive I'd say that it was a fairly decent charge as I've personally paid a lot more for an Uber back home.
The accomodation in Old Phuket Town
The cleanliness and hospitality of the Thai people are second to none so you could really get away with a 2-star hotel and have your basic luxuries of aircon and wi-fi. We always make our bookings via bookings.com just because it's so convenient you can chop and change your bookings without cost as you develop your itinerary. We chose to stay at The Rommanee Classic Guesthouse. One night here will set you back THB712 / ZAR286 or $9. You might think it is too cheap to be true so check them out here for yourself https://www.booki.ng/2fiNpK9. We booked a double room for the night and this hotel is so central you'll be walking distance from just about everything in the area.
Getting around or to Rassada Pier
The staff at the hotel are super helpful with this and will arrange a taxi for you on your request. To get down to Rassada pier we paid THB250 / ZAR100 or $7.50 the car was a new SUV and I think that it was a very reasonable fee and the car was clean and not once did we feel unsafe. If you would like a second point of view check out my wife's blog 
Peace be the journey.