What is the angle between the noon sun on the winter solstice and the southern horizon for an observer at a latitude of 38 N?

At the winter solstice for the northern hemisphere, the Sun is over the Tropic of Capricorn at 23 degrees 26 minutes south. Your observer is at 38N, a difference in zenith angle of 38+23.5=61.5 degrees. The elevation of the Sun above the southern horizon at noon (more precisely "Local Apparent Noon") is 90 - the azimuth angle, or 28.5 degrees.

Noon by your clock probably isn't "local apparent noon", the time when the Sun is in transit - at its highest point in the sky. For example, I live near Sacramento, CA, and here, in daylight savings time, the Sun transited my longitude at 12:57PM today. If I were a sailor (which I used to be) I could take a single "noon sight" of the Sun and calculate my position quite accurately with a single observation.

I need to watch the Sun through my sextant, and take continuous readings of the elevation angle of the Sun. (On the deck of a pitching boat, this observation can be tricky!) I note the time when the Sun is no longer rising in the sky, and starts to set; that time, to the nearest 15 seconds, gives me my longitude. The elevation angle at that time tells me my latitude. I do need to have an accurate watch, and a copy of the Nautical Almanac, but every modern sailor has these things.

Back in 1714, the British Admiralty offered a prize of twenty thousand pounds - a king's ransom, back then! - for the invention of a clock that kept accurate time at sea, because your celestial position is only as accurate as your clock!