Is it physically possible to see mountains from 500 km away? Quoting from lecture notes of Andrew young at San Diego State University:


  For less extreme, but very reliable, observations, consider some listed by Commander C. L. Garner of the Coast and Geodetic Survey in 1933 [...] He also credits the 1911 sighting of the Fairweather Mountains in Alaska [highest point is Mt. Fairweather 15 325 ft or 4670 m] from the ship Explorer from the Gulf of Alaska, 330 miles [531 km] away."


Earth’s mean average radius is 3959 miles by a right angled tangent of 330 miles gives a hypotenuse extending well above the surface of the earth of sqrt(39592 + 3302) = 3973 miles. Subtracting the radius 3959 miles gives the drop in the curvature of the earth from the beginning point where the Explorer made the sighting: 13.7 miles, or 72 492 ft. 

72 492 ft minus Mt. Fairweather’s height 15 325 ft gives 57 167 ft to spare for a whole mountain range to disappear beneath the curvature of the earth.  

What could account for this sighting 'through' the curve of the earth?
I think what was observed is a superior mirage. A superior mirage occurs when warm air overlies a very cold layer of air, aka an inversion. You don't often see a superior mirage in lower latitudes, but they seem to be somewhat common in and near the Arctic regions.With normal atmospheric conditions and taking into consideration refraction, about the furthest one could see a 15,325 foot mountain from, say, a 100 foot high deck is roughly 177 miles, depending on the tide. The only possible explanation is a very strong superior mirage.There are a couple of good articles here and here that describe the conditions necessary: