Since July 2019, there is an e-Visa valid for Kaliningrad, available to Austrians amongst others. The application website mentions two possible rail border crossings, one in Mamonovo and one in Sovetsk. RZD mentions restrictions on their website concerning rail travel to Kaliningrad and Leningrad oblasts when on e-Visa, and warns of some specific checkpoints where the e-Visa is invalid (Buslovskaya, Vyborg, Ivangorod, Nesterov).
Does anybody know whether there is a train connection crossing the Sovetsk border checkpoint currently, and if it actually is possible to take it with the e-Visa?
I was not able to find one with RZD or other train providers in Russia, but I have found that there seem to have been connections from Klaipeda at some point.
Alternatively, if there are none, is it possible to take one of the trains on the Vilnius-Kaliningrad route via the Kybartai checkpoint? It is not listed as allowed by KDMID, but not disallowed by RZD either.
A friend of mine took the train via Kybartai in June 2019, with the European Games exception, and could cross successfully, with the guards taking their time to check his ticket, and mentioning something about "Futbol".
Train from EU to Kaliningrad seems very related, but from the time before e-Visas were introduced.