In the closing credits the copyright date is MCMXLIII which is 1943, the correct Roman number is MCMLXIII. The X (10) is on the left of the L (50) so it is subtracted making the number 40. Placed to the right of the L (50) the X (10) is added giving 60 .
The Martian mentioned a Russian Astronaut, Russia does not use that terminology. Russia has Cosmonauts.
Tim finds Martin and takes him home where Martin convinces him not to write a story about him, but a detailed story appears in the newspaper. Tim had no opportunity to write a story even if he wanted to. Not a goof: Tim called the newspaper and told to story to someone in Rewrite. The story was about the space launch, not about the Martian.
In the very first scene, Tim's alarm clock goes off at 8:00, but is set to go off at 7:00.