Jutro muszę oddać książkę do biblioteki.

Because książkę is the accusative singular form of książka.

In this sentence, książkę is the direct object of oddać (to return / give back), and direct objects often take the accusative in Polish.

• książka = nominative, the basic dictionary form
• książkę = accusative singular

This is a very common pattern with feminine nouns ending in -a:

• kawa → kawę
• herbata → herbatę
• książka → książkę

Because do means to / into, and it requires the genitive case.

So:

• biblioteka = nominative
• do biblioteki = to the library / back to the library

Here, the phrase shows the destination of the action: where the book must be returned.

Compare:

• do biblioteki = to the library (movement toward)
• w bibliotece = in the library (location)

So:

• Oddać książkę do biblioteki = return the book to the library
• Być w bibliotece = be in the library

Oddać means to give back, to return, or sometimes to hand in, depending on context.

In this sentence, it means to return a book to the library.

A useful idea is that oddać is a perfective verb. That means it focuses on the action as completed.

So:

• oddać = return it, give it back, complete the act
• oddawać = be returning / return in a general or repeated sense

Because the sentence is about a specific thing that has to be done tomorrow, oddać is the natural choice.

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

• muszę = I must / I have to

So ja is not necessary.

You could say:

• Jutro muszę oddać książkę do biblioteki.

or

• Ja jutro muszę oddać książkę do biblioteki.

But adding ja usually gives extra emphasis, something like:

• I have to return the book tomorrow.

Most of the time, the version without ja sounds more natural.

Because in Polish, the present tense of a modal verb like musieć can describe a future obligation when the time is made clear by a word like jutro.

So:

• muszę literally looks like I must
• but with jutro, the meaning becomes I have to tomorrow / I must tomorrow

The future time is understood from jutro.

This is very natural in Polish and similar to English in sentences like:

• Tomorrow I have to return the book.

Because Polish word order is fairly flexible, and putting jutro first emphasizes the time.

• Jutro muszę oddać książkę do biblioteki. = Tomorrow I have to return the book to the library.
• Muszę jutro oddać książkę do biblioteki. = also correct
• Książkę muszę jutro oddać do biblioteki. = also possible, with different emphasis

Beginning the sentence with jutro is very common when the time is important or when setting the scene first.

Yes, there can be.

• muszę oddać = I must / I have to return
• mam oddać = can mean I am supposed to return or I am to return

Muszę usually sounds like a stronger personal obligation or necessity.

Mam oddać can sound more like:

• an instruction,
• an arrangement,
• or something expected by someone else.

In your sentence, muszę oddać is the most straightforward way to say that returning the book is necessary.

Because Polish has no articles like a / an / the.

So:

• książka can mean a book or the book
• biblioteka can mean a library or the library

The exact meaning is understood from context.

In this sentence, English naturally uses the because both speaker and listener probably know which book and which library are meant. Polish does not need a separate word for that.

Usually, in this context, it means return a library book.

Because of the context and the common phrase, a Polish speaker will normally understand:

• I have to return the book to the library tomorrow.

If you wanted to clearly say donate a book to the library, Polish would usually use a more specific expression, for example a verb related to donating or giving rather than the standard oddawać/oddać used for returning borrowed things.

So the most natural reading here is return.

A rough guide:

• jutro ≈ YOO-troh
• muszę ≈ MOO-sheh
• oddać ≈ OD-dach (the final ć is a soft ch-like sound)
• książkę ≈ roughly KSHYONSH-keh
• biblioteki ≈ bee-blee-oh-TEH-kee

A few useful points:

• sz sounds like English sh
• j sounds like English y
• ą and ę are nasal vowels, though in real speech their pronunciation can vary depending on what follows
• ć is a soft consonant, somewhat like a very soft ch

The hardest word here for many learners is usually książkę.

Yes, it is completely natural.

Jutro muszę oddać książkę do biblioteki. is a normal everyday sentence and sounds exactly like something a native speaker would say.

You might also hear:

• Muszę jutro oddać książkę do biblioteki.
• Jutro muszę zwrócić książkę do biblioteki.

But your original sentence is perfectly correct and idiomatic.

The sentence follows this structure:

• Jutro = time expression
• muszę = conjugated modal verb, I must / I have to
• oddać = infinitive
• książkę = direct object in the accusative
• do biblioteki = destination phrase with do + genitive

So the pattern is roughly:

time + modal verb + infinitive + object + destination

A literal breakdown would be:

• Jutro = tomorrow
• muszę = I must
• oddać = return / give back
• książkę = book
• do biblioteki = to the library

This is a very useful sentence pattern in Polish.