If it's a Mercator projection, then

long = x
lat = atan(sinhy)

Where lat, long, x, y of the center point == 0.

Most probably.

This should be adjusted to scale, of course. This can be done for long, as it's just linear. Lat is a bit harder, however. You should find a point with known lat_real on your map, then find it's lat_formula using the above formula. If the map was linearly stretched (which it probably was), you can use

lat = (lat_real / lat_formula) * atan(sinhy)

I strongly advise you to check these formulas for sanity before using them, though (just check several more points, like cities, peninsulas or whatever is visible on your map).