The newspaper Constantine picks up (after being transported back in time) is dated Friday, January 16, 1969. (1) January 16, 1969 was a Thursday. (2) The headline was "Man Walks on Moon" with a picture of Neil Armstrong on the moon. The moon landing was July 20, 1969 at 20:17 UTC. The whole mission was July 16-24, 1969--with launch on July 16, 1969 at 08:32 est.
For the scene in the coffee shop they used the set for CC Jitters often seen in The Flash (2014) which is in the same universe. Prior to the scene, the title in says "Star City 2017" and there is a table advertisement labeled "SC Jitters" but you can see out the front door a newspaper rack for Central City Picture news and "CC Jitters" is on the mugs.