England fans heading over for the last group game against Belgium, it is not anything that a budget flight to Gdansk and a quick bus ride cannot solve. Kaliningrad is the sole Russian Baltic Sea port that's ice-free all year round and hence has an important part in maintenance of the Baltic Fleet. Following your city tour you're totally free to explore the city independently provided that you don't leave St Petersburg.
Foreigners entering Russia is going to be fingerprinted. An e-visa doesn't allow travel to other areas of Russia. It will be essential to apply for a different form of visa so as to travel to other areas of Russia.