The Geographic North Pole or Terrestrial North Pole, is defined as the point in the Northern Hemisphere where the Earth's axis of rotation meets its surface. source
There were several attempts to reach the North pole prior to the 1940's. A review of Wikipedia's article on Exploration of the North_Pole indicate some of the difficulties in definitively identifying the location.
I imagine the best time is either near summer when the sun can be your best resource or the winter when the stars can be your best resource. But I have no idea which of either of these times would be best.
Excluding the use of GPS, and just using tools available to earlier explores what time of year is best to find the North Pole. Ignoring all the difficulties of travel at that time of year, just identifying the location with pre-GPS technology.