As some already know there will be a number of special trains to the World Cup Host Cities that fans will be able to book for FREE. That option includes Kaliningrad as well.
You can check the schedule already.
Unfortunately Moscow is the only city available so far. I as hoping to visit Saint Petersburg for free.
But note that the schedule is not final and shall be subject to change. It will be finalized after December 15, 2017.
Also important to know that by taking such train you will cross borders of Lithuania and Belarus. No information is available if there will be any special visa-free solutions available. Check the requirements of those countries for your citizenship, what is needed to cross the border for you.
p.s. And yeah, this is post is about FREE Trains. At the same time there are a lot of regular trains and plains from Kaliningrad to major Russian cities and several neighbor countries.
Feel free to ask any questions regarding visiting Kaliningrad and Russia for the FIFA World Cup 2018.