I was wondering if it were possible to extract coordinates from a static map picture like this:
Is it possible to extract the coordinates of the routes? The only idea I can come up with, other than manually getting them by hand, is by overlaying the map and extracting the exact coordinates that way.
The process that you are looking for is called georeferencing in a GIS context.
In order to determine the latitude/longitude coordinates of a point (or series of points in a line), you need to first establish coordinates of other known points. These are reference points of known locations (such as a distinctive coastline, or a city). Applying these to the raster image that you have, you can then overlay it on a map in a GIS application and then query other unknown locations on the image (such as the routes) to establish their latitude/longitude.
You could attempt this in a graphics program by looking at the x/y coordinates of the route pixels and compare to a known reference point pixel; however, the math on that is going to be tedious and you also wouldn't account for the map projection. Both of those are taken care of by georeferencing.
I would note that you should consider the results you get, even in a well-referenced GIS, to be approximate. The map deals with a very large spatial area, and the routes cover long distances. But since it dates from 2012 it was presumably made in a GIS application and so at least the source data is likely to be accurate :)
Additional resources: