Do any of the other Professor Layton games contain information that impacts on this story? Such as who characters are.
Professor Layton and the Last Specter (or the Spectre's Call in Europe) is actually a prequel to the first three games in the series. I've played the first three games to completion, (this game I've made some progress on but haven't finished) and there's some minor plot overlap, (a few shared secondary characters, for the most part) but there's enough backstory in all of the games that you can pick them up out of order and not get too lost.
In fact, the first game kind of drops you into the middle of Layton's story anyhow, and fills in the relevant details as it goes along. The primary antagonist of that game doesn't really get a proper backstory (and a reason to hate Layton) until game #3, for example.
The focus of the game is really on the puzzles anyhow, and the plot itself tends to be more or less nonsensical for 80% of the game, so I wouldn't stress playing them out of order, and especially not if you start with Spectre's Call first.