Why late arrivals need planning here
Two things combine. The airport is at the northern end of a large island and most visitors are staying somewhere considerably south of it, so the transfer is a real drive rather than a hop. And there is no published fare structure to fall back on, which means the price is whatever is agreed at the moment you agree it.
At two in the afternoon that is a mild annoyance. At one in the morning, after a long-haul flight, with children or a lot of luggage, it is a situation where you have very little leverage and no basis for comparison.
Book it before you fly
Message your accommodation with your flight number, not just your arrival time, and ask two things: what they charge for an airport transfer, and whether reception will be staffed when you arrive. Most Phuket properties arrange transfers as routine.
Having a flight number matters because it lets a driver track a delay. An arrival time in an email is a guess; a flight number is information.
If you have not pre-booked
Use the counters inside the terminal rather than accepting an offer outside. Airports of Thailand lists limousine counters in the Domestic terminal on Floor 1 in the arrivals public area, close to both arrival halls, with sedans, SUVs and minibuses available, and lists taxi and shuttle bus services alongside them.
Agree the destination and the price at the counter, indoors, before you walk out with your bags. Have your accommodation's name, area and phone number ready to show, since Phuket has multiple similarly named properties spread across very different parts of the island.
What to have ready before you land
Consider not crossing the island tonight
If you land after midnight and your hotel is at the far south of the island, there is a reasonable case for spending the first night near the airport at Mai Khao or Nai Yang and making the drive in daylight. It costs one night's accommodation and buys you an arrival that is not a two-hour ordeal at the end of a long flight.
This is particularly worth considering with small children, with a lot of luggage, or when the alternative is arriving somewhere unfamiliar in the dark with nobody at the desk.