If I'm not mistaken they had to skip a couple of days somewhere around 1500 because they had been wrong for ages. But since when did they know a year takes approximately 365 days and how did they discover it?
I can't answer the main part of your question, but it may be worth pointing out a misapprehension: the introduction of the Gregorian calendar in 1582 wasn't to correct the year length to 365 days, it was to correct the length from 365.25 days (Julian calendar) to 365.2425 days. The actual length of a solar year is close to 365.2422 days.
The Julian calendar (365.25 days) was introduced in Rome in 46 BC. But as I say I can't answer the main part of your question about when the length of the solar year was first known.