Learn Hardcore Greek: Το μουσείο στην πόλη είναι δωρεάν την Κυριακή. - The museum in the city is free on Sunday.

Greek articles must match the noun in gender, number, and case. Since μουσείο is a neuter singular subject, you need Το in front of it: Το μουσείο = the museum.

What gender and type of noun is μουσείο, and does its form change?

μουσείο is a neuter noun. It typically belongs to the neuter class ending in ‑ο / ‑ιο.

Basic forms of μουσείο:

το μουσείο – the museum (nominative/accusative singular)
του μουσείου – of the museum (genitive singular)
τα μουσεία – the museums (nominative/accusative plural)
των μουσείων – of the museums (genitive plural)

In this sentence it is the subject, so it appears as το μουσείο (neuter nominative singular).

What exactly is στην? Is it one word or a combination?

στην is a contracted form of the preposition σε (in, at, to) plus the feminine definite article την (the):

σε
- την → στην

It is used before a feminine singular noun in the accusative case, like πόλη:

στην πόλη = in the city / in town

You will also see:

στον = σε
- τον (masculine)
στο = σε
- το (neuter)

Why is πόλη in this form, and what case is it?

πόλη is a feminine noun meaning city / town. In στην πόλη, it is in the accusative singular case, because:

The preposition σε (in, at, to) is followed by the accusative.
την is the feminine accusative singular article.
The noun appears as πόλη in both nominative and accusative singular, so the ending looks the same.

So στην πόλη literally means in the city (with πόλη functioning as the object of the preposition).

Why is it στην πόλη and not just σε πόλη?

Both are grammatically possible, but they differ in meaning:

στην πόλη = in the city / in town (a specific city that speaker and listener can identify from context)
σε πόλη = in a city (nonspecific, any city)

In this sentence, στην πόλη suggests a particular town/city that is already known or obvious from context, roughly like saying in the city or in town in English.

What form of the verb is είναι, and what does it correspond to in English?

είναι is the present tense, indicative mood of the verb είμαι (to be). It serves as both:

3rd person singular: he/she/it is
3rd person plural: they are

Context tells you which one is meant. Here the subject is Το μουσείο (singular), so είναι = is.

The full present tense of είμαι:

εγώ είμαι – I am
εσύ είσαι – you are (singular)
αυτός/αυτή/αυτό είναι – he/she/it is
εμείς είμαστε – we are
εσείς είστε – you are (plural or polite)
αυτοί/αυτές/αυτά είναι – they are

What kind of word is δωρεάν? Does it change form for gender or number?

δωρεάν means free (of charge), for free. Grammatically, it behaves like an indeclinable adjective/adverb:

It does not change for gender, number, or case.
You use the same form with masculine, feminine, and neuter nouns, and also in an adverbial sense.

Examples:

Το μουσείο είναι δωρεάν. – The museum is free.
Η είσοδος είναι δωρεάν. – The entrance is free.
Μπαίνεις δωρεάν. – You enter for free.

So in Το μουσείο … είναι δωρεάν, δωρεάν is functioning like the English predicate adjective free but keeps one fixed form.

Why do we say την Κυριακή instead of just Κυριακή for on Sunday?

Greek normally uses the definite article with days of the week when saying on [day]:

την Δευτέρα – on Monday
την Τρίτη – on Tuesday
την Κυριακή – on Sunday

So την Κυριακή = on Sunday.

Without the article, just Κυριακή, it typically functions as the name of the day (Sunday) rather than the time expression on Sunday, especially in isolation. In everyday speech, people may drop the article in short answers, but in a full sentence like this, την Κυριακή is the normal form.

Does την Κυριακή mean this Sunday, on Sundays (in general), or both?

την Κυριακή can be context‑dependent:

A specific Sunday (this/that one)
- If you are talking about one particular Sunday (coming or past):
  - Το μουσείο στην πόλη είναι δωρεάν την Κυριακή.
    = The museum in the city is free this/that Sunday.
A regular habit: on Sundays in general
- Very often, especially in rules, schedules, or general facts, την Κυριακή is understood as on Sundays (regularly):
  - Την Κυριακή τα μαγαζιά είναι κλειστά.
    = On Sunday / On Sundays the shops are closed.

To make every Sunday completely clear and explicit, Greek also uses:

κάθε Κυριακή – every Sunday
τις Κυριακές – on Sundays (plural)

But in many real-world sentences like this one, την Κυριακή will naturally be read as a habitual fact: on Sundays the museum is free, unless context points to a single Sunday.

Why is Κυριακή capitalized here? Are days of the week usually written with a capital letter in Greek?

Strictly speaking, in standard modern Greek spelling, days of the week are usually written with a lowercase initial letter, unless they start a sentence:

η Κυριακή at the beginning of a sentence → Η Κυριακή
elsewhere: την Κυριακή, κάθε Κυριακή

In practice, you will often see them capitalized (in signs, textbooks, etc.), imitating English. So:

την Κυριακή (lowercase κ) is the usual modern rule.
την Κυριακή (with capital Κ) is also common in informal or mixed usage, and learners see it a lot.

The key point is linguistic, not orthographic: the article την is what turns it into on Sunday.

Can the word order change, for example to Την Κυριακή το μουσείο στην πόλη είναι δωρεάν? Is that still correct?

Yes. Greek word order is freer than English. All of these are grammatically correct:

Το μουσείο στην πόλη είναι δωρεάν την Κυριακή.
Την Κυριακή το μουσείο στην πόλη είναι δωρεάν.
Στην πόλη το μουσείο είναι δωρεάν την Κυριακή.

The basic information stays the same; changing the order mostly affects emphasis:

Starting with Την Κυριακή highlights the time:
Την Κυριακή το μουσείο στην πόλη είναι δωρεάν.
= As for Sunday, on that day the museum in the city is free.
Starting with Στην πόλη highlights the place.

English allows some movement but is more rigid; Greek relies more on articles and endings than on word order for grammar.

How do you pronounce Το μουσείο στην πόλη είναι δωρεάν την Κυριακή?

Approximate pronunciation with stress marked in capital letters:

to mu-SI-o stin PO-li I-ne tho-re-AN tin kiri-a-KI

IPA (one common transcription):

[to muˈsio stin ˈpoli ˈine ðoɾeˈan tin ciriaˈci]

Notes:

ου in μου- is like oo in food.
ει / ι / η are all pronounced like ee in see.
θ (as in δωρεάν – careful, that one starts with δ) is like th in think;
δ is like th in this.
κ before ι (as in Κυριακή) is a bit softer than English k, but kee-ree-a-KEE is close enough for learners.

What is the difference between Το μουσείο στην πόλη and Το μουσείο της πόλης in meaning?

Both involve the museum and the city, but they are not the same:

Το μουσείο στην πόλη
= The museum in the city / the museum that is located in town.
Focus: location.
It just states where the museum is.
Το μουσείο της πόλης
= The city’s museum / the museum of the city.
Focus: possession/affiliation.
It presents this museum as the official or main museum belonging to or associated with the city.

In your sentence, Το μουσείο στην πόλη means simply the museum that is in town, not necessarily the official museum of that city.