Any ideas anyone?
Take a train from Gdańsk to Olsztyn, then from Olsztyn to Braniewo. From there, get a taxi to the Russian border. You can't walk across the border, but just look for someone willing to let you come with them across the border for some cash (20zł should do it). From the other side, you should be able to agree a lift into Kaliningrad with someone.
edit: just seen that you want to do it in reverse. In that case, get a taxi from Kaliningrad city to the border, then just get someone to take you across, then train from Braniewo to Gdańsk.
There may be some train from Kaliningrad to Mamonovo (next to the border), but I'm not sure.
edit 2: Yeah, there's a train from Kaliningrad to Mamonovo, three per day. You can try that, but it'll probably be easier to just get a taxi to take you straight to the border - should be no more than 25-30 Euro.