Mam wolne popołudnie.

Because Polish uses the verb mieć (to have) here, just like English does in I have a free afternoon.

• mam = I have
• jestem = I am

So:

• Mam wolne popołudnie = I have a free afternoon
• Jestem wolny / wolna po południu would mean something more like I am free in the afternoon

These are related ideas, but not the same structure.

In Polish, subject pronouns are often omitted because the verb ending already shows who is doing the action.

• mam clearly means I have
• so ja (I) is not necessary

You could say Ja mam wolne popołudnie, but that usually adds emphasis, for example:

• Ja mam wolne popołudnie, ale on pracuje.
  I have a free afternoon, but he is working.

Normally, Mam wolne popołudnie is the natural choice.

It is in the accusative, because it is the direct object of mam.

The verb mieć (to have) usually takes the accusative:

• Mam czas = I have time
• Mam książkę = I have a book
• Mam wolne popołudnie = I have a free afternoon

However, popołudnie looks the same in the nominative and accusative. That is common for many neuter singular nouns in Polish.

So:

• nominative: popołudnie
• accusative: popołudnie

Same form, different grammatical role.

Because wolne has to agree with popołudnie.

The noun popołudnie is:

• neuter
• singular
• here, in the accusative

The adjective must match it, so we get:

• wolny for masculine
• wolna for feminine
• wolne for neuter

Examples:

• wolny dzień = a free day / a day off
• wolna sobota = a free Saturday
• wolne popołudnie = a free afternoon

Since popołudnie is neuter, wolne is the correct form.

Here wolne means free, unoccupied, or off.

It does not mean politically free or abstract freedom here. It means that the afternoon is not filled with work, duties, or plans.

Common meanings of wolny / wolna / wolne include:

• free, available
• not busy
• off work / having time off
• empty, unoccupied
• slow, in some contexts

In this sentence, the idea is:

• I have a free afternoon
• My afternoon is free
• I’m off this afternoon, depending on context

Yes. Mam wolne is a very common and natural expression.

It often means:

• I’m off
• I have time off
• I don’t have work/classes today
• I’m free

The exact meaning depends on context.

Examples:

• Dziś mam wolne. = I’m off today.
• Jutro mam wolne. = I have tomorrow off.
• Po południu mam wolne. = I’m free in the afternoon.

So Mam wolne popołudnie is basically a more specific version of that pattern.

This is a very useful distinction.

• popołudnie = afternoon as a noun
• po południu = in the afternoon / after noon as a time expression

So:

• Mam wolne popołudnie = I have a free afternoon
• Mam wolne po południu = I’m free in the afternoon

The first talks about the afternoon itself as a thing you have.
The second talks about when you are free.

Both are natural, but they are not identical.

Yes, Polish word order is fairly flexible, but the neutral order here is:

• Mam wolne popołudnie.

This is the most straightforward, natural version.

Other orders are possible, but they change emphasis:

• Wolne popołudnie mam. — marked, contrastive, less neutral
• Popołudnie mam wolne. — emphasizes the afternoon
• Mam popołudnie wolne. — understandable, but less natural

For a learner, the safest default is:

• Mam wolne popołudnie.

Approximately:

po-po-WOOD-nyeh

A more careful breakdown:

• po = like po in a short poh
• second po = same
• ł sounds like English w
• u sounds like oo
• dnie sounds roughly like d-nye

Stress in Polish usually falls on the second-to-last syllable, so here the stress is on:

• po-po-ŁUD-nie

A rough pronunciation guide:

• popołudnie ≈ poh-poh-WOOD-nyeh

Yes, popołudnie is a neuter noun.

One clue is the ending -e, which is often associated with neuter nouns, though not always. Another clue is dictionary information: nouns are learned together with their gender.

Because it is neuter, adjectives with it take neuter forms:

• długie popołudnie = a long afternoon
• spokojne popołudnie = a calm afternoon
• wolne popołudnie = a free afternoon

So if you memorize popołudnie as a neuter noun, the adjective form becomes easier to understand.

Yes, this is natural Polish.

It sounds like something someone might say when talking about plans, availability, or time off.

For example:

• Dziś mam wolne popołudnie. = I have a free afternoon today.
• W końcu mam wolne popołudnie. = I finally have a free afternoon.
• Masz wolne popołudnie? = Do you have a free afternoon?

Depending on context, a Polish speaker might also choose:

• Mam wolne po południu. = I’m free in the afternoon.
• Mam dziś wolne. = I’m off today.

But your sentence is absolutely normal and useful.