To understand the Russian psyche, you could try reading Turgenev and Tolstoy, sit through plays by Gogol or Chekhov, or watch the Tchaikovsky/Petipa ballet of Swan Lake.
A simpler method might be to see a film by Andrei Tarkovsky, a director of quintessentially Russian cinema and particularly Russian sci-fi. The original movie version of Solaris, made by Tarkovsky in 1972, is currently available on channel4.com.
The 2010 remake by Steven Soderbergh, with George Clooney and Natascha McElhone, was compellingly eerie in its own way but its brand of strangeness was quite different from Tarkovsky's, which is subtle, slow-moving and hard to define. For one thing, it is more willing to play with the conventions of narrative. In this, and in its sense of detachment, it seems to echo Kubrick's 2001, made four years before Solaris — though Tarkovsky was contemptuous of 2001 and clearly hoped to do better. Solaris has not aged as well as 2001 but remains the more psychologically interesting film.
Similar as they are in tone, the contrast between these two movies from the hippie era — Kubrick's and Tarkovsky's — could be said to highlight differences between American and Russian mentality. (One can throw in a third film from the same genre/era to provide the French perspective: Jean-Luc Godard's Alphaville.)
Even better than Solaris is Tarkovsky's 1979 sci-fi movie Stalker (the title is somewhat misleading), in which the director shows his genius for generating atmosphere and philosophical reflection, in spite of there being relatively little dialogue or plot.