There should be van/mini-bus service also to Kata Beach. Expect to pay around 200-250 B/person. For 2 persons with luggage I wouldn't bother. Not worth the hassle, extra time (waiting for the van to be full/depart and making stops enroute) when 2 persons and arriving around midnight. You risk to wait very long until the van will depart at that hour. 1,600 B for a return transfer to Karon is very reasonable; book it! If you book at the airport expect around 1,000 B to Kata (one way!).
Unfortunately, in Phuket taxi and Grab doesn't work as you are used to in BKK or elsewhere in Asia. If you expect significant savings by using Grab or taxi compared to the inflated fixed rates dictated by the local taxi mafia, you will be disappointed.