First, two things. make sure you have Steam automatically update the game for you. This a common mistake users make and also make sure you are not in Offline mode.
To make sure it's set to automatically update:
In the left hand side window were all your games are stored, right click on the name of the game, in this instance it will be Football Manager 2013, go to Properties and then the Updates tab. If it says always keep this game updated then you are fine on that.
One thing to note: When starting a new game, the database will still say 13.0.0 even though its the newest update. So don't get confused with this if this is the case.