The last person to fly solo in space was Brian Binnie of the Space Ship One flight on October 4, 2004. That was a sub-orbital flight, just barely touching space. The Space Shuttle required 2 people to launch minimum, thus, it was never launched Solo. Neither were Apollo and Gemini, so it has been since the Mercury program since an American could launch solo in to space.
Of some note is the fact that, while launched together, the Command Module pilot was in space by himself for an extended period of time. Ronald E Evans was alone on the Command Module for 80 hours on Apollo 17. The Russians, however, orbited Earth for slightly longer solo.
The Chinese have flown one solo mission, and the pilot was Yang Liwei. The date was October 15, 2003. He was the last astronaut to orbit the Earth solo.
The last solo Russian Cosmonaut was Georgy Beregovoy, launched 30 October 1968. Since then, the Russian program has used the Soyuz capsule, which typically requires two pilots. Soyuz 4/5, however, transferred crew in space. Soyuz 4, launched 17 January 1969, was launched with just Vladimir Shatalov, while Soyuz 5 landed with just Boris Volynov, having two astronauts from Soyuz 5 who switched in space. This was the last time a Russian spacecraft had only a single occupant in space, per Wikipedia.
Of course, the movie was filmed in 1983. That leaves Georgy Beregovoy as the last person to fly an entire mission solo in space prior to the movie being released.