How long does a specific moon last during Superbrothers: Sword & Sworcery EP?

It lasts the same length as the specific moon phase lasts in real life give or take a bit of approximation by the game. The average lunar cycle lasts about 29.5 days and there are considered to be 8 main moon phases, new, waxing crescent, first quarter, waxing gibbous, full, waning gibbous, last quarter, and waning crescent. While not precise because some phases last slightly different durations than others depending on the positions of the moon, the earth, the sun, and the person observing the moon, rather than complicated astronomical math, a simple approximation is about 3.7 days/phase or so (29.5 days/8 phases). See any site on lunar phases and the lunar calendar that you like, but for simplicity, here's the relevant source information on the wikipedia article.

I'm not sure if the game factors in eclipses.

You can temporarily change the moon phase in the game if you'd rather not wait by going to

the moon grotto

or you can just change your system date if you don't care about only getting 99% completion.