W upalne popołudnie szukam cienia.

Because this is a time expression, not a location expression.

With w, Polish often uses:

• locative for location: w domu, w lesie
• accusative in some expressions meaning on / during / in a period of time: w poniedziałek, w przyszły weekend, w upalne popołudnie

So w upalne popołudnie means on a hot afternoon / during a hot afternoon.

It is in the accusative singular.

The preposition w here requires the accusative because the phrase is being used as a time expression.

Also, popołudnie is a neuter noun, and for many neuter nouns the nominative and accusative forms look the same, so:

• nominative: upalne popołudnie
• accusative: upalne popołudnie

Because upalne has to agree with popołudnie.

Popołudnie is:

• neuter
• singular
• here, accusative

So the adjective must match it:

• upalne popołudnie

This is standard Polish adjective-noun agreement.

It is neuter.

A useful clue is that many Polish nouns ending in -o or -e are neuter, and popołudnie is one of them.

That is why you get forms like:

• to popołudnie
• upalne popołudnie

Because Polish usually omits subject pronouns when they are already clear from the verb ending.

Szukam means I am looking for / I look for, so ja is not necessary.

You could say Ja szukam cienia, but that would usually add emphasis, for example:

• Ja szukam cienia, a on szuka słońca.

Szukam is the 1st person singular present tense form of szukać.

So:

• szukać = to look for
• szukam = I look for / I am looking for

This verb is imperfective, so it is the normal choice for an ongoing, repeated, or general action.

Because szukać normally takes the genitive, not the accusative.

So:

• nominative: cień
• genitive: cienia

After szukać, you ask:

• szukać kogo? czego? = to look for whom? what?

So:

• szukam cienia
• not szukam cień

This is a very common thing learners have to memorize with szukać.

It can mean both, depending on context.

In this sentence, it clearly means shade, because the context is a hot afternoon and someone wants relief from the sun.

So even if you already know cień as shadow, here the natural English meaning is shade.

Yes. Polish word order is fairly flexible.

The original sentence:

• W upalne popołudnie szukam cienia.

puts the time expression first, which sets the scene.

You could also say:

• Szukam cienia w upalne popołudnie.

That is still correct, but the focus feels a little different.

If you move cienia to the front, it sounds more contrastive or stylistically marked:

• Cienia szukam w upalne popołudnie.

So the original version is a very natural neutral sentence.

Because Polish has no articles.

So Polish does not have separate words corresponding directly to English a, an, or the.

Whether something is definite or indefinite is usually understood from:

• context
• word order
• emphasis
• the situation

So cienia can mean shade or the shade, depending on context.

Not exactly.

• gorący is a general word for hot
• upalny is more specifically hot/sweltering, especially about weather

So upalne popołudnie sounds very natural for a hot, oppressive afternoon.

You could understand gorące popołudnie, but upalne is the more idiomatic choice here.

A rough pronunciation for an English speaker is:

v oo-PAL-ne po-po-WOOD-nye SHOO-kam CHEN-ya

A few useful points:

• w is pronounced like English v
• sz sounds like English sh
• cie in cienia is a soft sound, roughly like chye
• ń is a soft n, similar to the ny sound in canyon
• popołudnie has stress on łud: po-po-ŁUD-nie

As usual in Polish, the stress is normally on the second-to-last syllable.