1. A time difference of 0.001 s correspond to a distance of 186 miles.
2. To determine the x-coordinate of the ship's position use the definition of the hyperbola as the locus of points whose difference of the distances to two fixed points is a constant. You have to solve for x:
$\displaystyle \sqrt{(-150-x)^2+75^2} - \sqrt{(150-x)^2+75^2} =186$
3. For confirmation only: I've got $\displaystyle x \approx 110.3$