According to the excellent official Kaliningrad tourist website with travel information:
you can travel to Kaliningrad by bus, plane, car , ferry (from Gdynia in Poland, not confirmed yet), train (from Russia only), bicycle and yacht :)
More detailed information will be provided here soon.
FOR YOUR TRAVEL PLANNING: a bus from GDANSK to KALININGRAD:
A good travel option might be take a flight/train or bus to GDANSK and change to the
direct bus GDANSK – KALININGRAD. Time: 3,5 hours, costs: 45 zloty (11 Euro)
there are 4 buses per day leaving at 6am, 9 am, 3 pm, 5 pm on all days of the week
from bus station: Dworzec PKS.Gdansk, ul. 3 Maja 12, Gdańsk (next to the main train station)
Here is the Timetable to download (look at “Czas letni”-column (summer time)
For a taxi from airport phone 587711222 (they speak English).
Costs for a taxi from the airport to the bus station (Dworzec PKS): 31 to 35 złoty (up to 9 Euro)