Fig.: to do things and make decisions by feelings and instinct rather than method or experience

My friend Jack has no sense of direction at all: if he drives by the seat of his pants, without looking at a map, he always goes the wrong way

Il mio amico Jack ha un senso di orientamento nullo; quando guida a lume di naso, senza guardare una cartina, prende sempre la strada sbagliata 

I know very little Spanish, so when I was in Mexico I just flew by the seat of my pants with mixed results, to put it mildly

La mia conoscenza dello spagnolo è veramente limitata, perciò quando ero in Messico sono andato a spanne con risultati a dir poco contrastanti

I don’t know exactly how much it costs, but off the top of my head I would say around $ 50

Non so esattamente quanto costi, ma così a naso direi sui 50 Euro

In the 1930s, early aviators could tell whether they were flying straight, ascending, or descending only by the pressure felt in their seats, which gave rise to the original meaning of this phrase. Similarly, pilots couldn't rely on instruments in low visibility situations, therefore they had to depend on their senses to determine their position.
Following the war, this expression was adopted for more general application.