The monument in the town of Bor, 250 miles east of Moscow was unveiled as Victory Day parades took place across Russia marking the end of World War Two.
President Trump will meet Putin on Monday in Helsinki, Finland.
The Russian leader is serving his second consecutive term, and the next presidential election he can enter will be in 2030.
Kazakhstan was one of the last countries to keep the controversial alphabet ordered by Russia.
"The current bastards in power are as insignificant as the ones then," one disgruntled critic tweeted.
The talk show's hosts and guests questioned Trump's understanding of the First Amendment and said whispers are spreading that he should be removed from office.
Ukraine's general prosecutor said it will charge Stalin and Lavrenti Beria over 70-year-old crimes, and is still investigating the mass starvation of 1932-33.
Independent polls have also found that nearly a majority of Russians find kind words to say about Stalin.
The Mask of Grief monument was built in 1996 at the site on the route to a labor camp.
"Stalin's face is at Arbatskaya, this is not a joke," Russians wrote on Twitter.
The Russian leader compared the Soviet code of ethics to the Bible.