Monoloğu ezberlemek üç günümü aldı.

The main finite verb is aldı.

• aldı = took
• ezberlemek = to memorize / memorizing

So the sentence is built around aldı, and monoloğu ezberlemek acts as the thing that took time.

A very literal breakdown is:

• Monoloğu ezberlemek = memorizing the monologue
• üç günümü = my three days / three days of my time
• aldı = took

So literally: Memorizing the monologue took my three days.

Because Turkish often uses the infinitive ending -mek / -mak to turn a verb into a noun-like expression.

• ezberlemek = to memorize / memorizing

Here, monoloğu ezberlemek functions like an action noun, similar to English memorizing the monologue.

So instead of saying something like I memorized the monologue, this sentence talks about the action itself:

• Monoloğu ezberlemek = the act of memorizing the monologue

That whole phrase is what took three days.

Because it is the direct object of ezberlemek, and it refers to a specific monologue.

In Turkish, a definite direct object usually takes the accusative ending.

• monolog = a monologue
• monoloğu = the monologue

So:

• monolog ezberlemek would sound more like to memorize a monologue
• monoloğu ezberlemek means to memorize the monologue, a specific one

Even though monoloğu is not the object of aldı, it is still the object inside the infinitive phrase monoloğu ezberlemek.

This happens because of two things:

1. The accusative suffix is added: -(y)ı / -(y)i / -(y)u / -(y)ü
2. The final g in some words softens before a vowel-initial suffix

So:

• monolog
  • -u → monoloğu

The final g changes to ğ when the vowel-initial suffix is added.

This kind of consonant softening is common in Turkish.

üç günümü literally means my three days or more naturally three days of my time.

Breakdown:

• gün = day
• günüm = my day
• günümü = my day, as a definite direct object
• üç günümü = my three days

In this sentence, Turkish expresses the idea as if the action took three days from your available time.

So:

• üç gün sürdü = it lasted three days
• üç günümü aldı = it took three days of my time

This is a very natural Turkish way to say it.

Because after numbers, Turkish nouns usually stay singular.

So:

• bir gün = one day
• üç gün = three days
• not üç günler

This rule still applies here, even though the noun also has possessive and accusative endings.

So:

• üç günüm = my three days
• üç günümü = my three days, as the object

Using günler after üç would sound wrong in standard Turkish here.

Turkish often expresses this idea differently from English.

English says:

• It took me three days

Turkish often says something closer to:

• It took three days of my time

That is why gün gets the first-person possessive ending:

• günüm = my day
• üç günüm = my three days

Then because this phrase is the object of aldı, it becomes:

• üç günümü aldı

So the me in English is not translated as a separate word like beni here. Instead, it is built into günümü.

Yes. almak literally means to take, and in this pattern it works very much like English take in expressions of time.

So:

• Bu iş iki saat aldı. = This job took two hours.
• Monoloğu ezberlemek üç günümü aldı. = Memorizing the monologue took me three days.

It is both literal and idiomatic. Turkish really does use take here.

No. üç günümü is the object of aldı.

Here is the structure:

• monoloğu ezberlemek = subject-like infinitive phrase
• üç günümü = direct object
• aldı = verb

So the action memorizing the monologue is the thing that performed the taking, and three days of my time is what got taken.

A rough structural translation is:

• [Memorizing the monologue] [took] [three days of my time].

The given order is natural and clear, but Turkish word order is somewhat flexible.

The original:

• Monoloğu ezberlemek üç günümü aldı.

This is a neutral way to say it.

You may also hear variations depending on emphasis, for example:

• Üç günümü monoloğu ezberlemek aldı.

But this sounds more marked and less neutral. In normal usage, the original order is much more natural.

Turkish often places the most important or focused information before the verb, so word order can change for emphasis, but the version you have is a good standard model.

Yes. A very common alternative is with sürmek:

• Monoloğu ezberlemek üç gün sürdü.
• Monoloğu ezberlemem üç gün sürdü.

These mean roughly Memorizing the monologue took three days or My memorizing the monologue lasted three days.

Difference in feel:

• üç günümü aldı emphasizes that it used up your time
• üç gün sürdü emphasizes the duration itself

Both are correct and natural, but üç günümü aldı has a slightly stronger sense of it took up three days of my time.