"Zimą powinieneś nosić czapkę."

Translation: In winter you should wear a cap.

[IE99]

Is this instrumental case? Why isnt the preposition "W" used here? I havent seen this sentence construction before.

[Jellei]

"W zimie" also works.

I'd say that it's not exactly Instrumental, it's an adverb that looks like the Instrumental version of the noun. Just like "wieczorem". And yes, I think I'd be more likely to say "Zimą" than "W zimie" in such a sentence.

The acceptable versions, according to this website (http://ulalupinska.pl/miniporadnia-zima-lato/) are: wiosną/na wiosnę, latem/w lecie, jesienią/na jesieni and zimą/w zimie.

[ADJD4]

Is there a reason why we would be 'in' summer or winter but 'at' autumn or spring?

[Jellei]

I wouldn't think about "na" as "at". In fact for a moment I was really confused, thinking that you're asking about English - why would you? :D

I don't know the reason. The website I linked also says that the author couldn't find any explanation 'why'. It is how it is, I'm afraid...

[Christian7652]

If that helps in slovak spring and fall are also na while summer and winter are v.so it has some protoslavic background i guess

[mph.vgc]

Why not - During winter you should wear a hat?

[Jellei]

OK, added.

[Larry11215]

you ought to = you should (but got nixed)

[Jellei]

Added "ought to".

[F4yY9kZj]

The (female) voice says 'powiniemyś' instead of 'powinieneś'.

[alik1989]

I don't think it does.