Determining your location when you have a map but not your starting location is a job for a particle filter.
(Wikipedia's entry on particle filters is not very helpful to beginners.) See also, this question.
Here's an animation of a particle filter in use. The black boxes are walls, the green turtle is the robot, the red marks are possible locations, and the grey circle is the center point of all the estimates. As the robot moves, the possible locations become more tightly grouped until there is no more ambiguity.
The idea is that by moving around the space and measuring the distance to walls around you (or by bumping into them), you can get a better sense of where you "might be".
For example: in the map you showed, imagine that your robot moves forward by 4 spaces but does not hit a wall. There are only a few spots on the map where this is possible, so now you know you must be in one of those places. The actual movement doesn't matter, but as you build up your knowledge of where the walls are (and where you don't find walls), some moves will be more useful in determining your location than others. And eventually, you'll have roughly 100% confidence in your location.