In the Bowery, one of the riddle answers is a plaque to commemorate the opening of Arkham City, the text on it reads:
November 19th 1###"
The rest of the date is missing due to the damaged plaque. But, considering that Arkham City was opened AFTER the first game, and a year and half BEFORE this game, does that mean the game takes place in the 20th century?
Batmans gear is a little advanced for that era, I mean, have you seen the stuff he's got.