المكتب مفتوح طيلة اليوم.

Arabic often makes present-tense statements without a word for is/are. This is called a nominal sentence.

So:

• المكتب = the office
• مفتوح = open

Together, المكتب مفتوح literally gives the office open, but it means the office is open.

If you wanted past or future, Arabic would usually use a verb, but in the present, it is often omitted.

It is a nominal sentence (جملة اسمية).

Its parts are:

• المكتب — the subject/topic (مبتدأ)
• مفتوح — the predicate/comment (خبر)
• طيلة اليوم — an expression of time/duration, meaning throughout the day / all day

So the basic structure is:

• المكتب = what we are talking about
• مفتوح = what we say about it
• طيلة اليوم = for how long

الـ is the Arabic definite article, meaning the.

So:

• مكتب = an office / office / a desk depending on context
• المكتب = the office

In this sentence, the noun is definite because we are talking about a specific office: the office is open all day.

Yes. مكتب can mean different things depending on context, such as:

• office
• desk
• sometimes bureau or agency

With الـ, المكتب could mean the office or the desk.
In this sentence, because of مفتوح طيلة اليوم, the natural meaning is the office.

Because مفتوح is the predicate of the sentence, not another definite noun.

In المكتب مفتوح:

• المكتب is definite: the office
• مفتوح describes its state: open

Arabic often uses an indefinite predicate after a definite subject in sentences like this.

So this pattern is very normal:

• الباب مغلق = the door is closed
• المحل مفتوح = the shop is open
• المكتب مفتوح = the office is open

It behaves like an adjective in meaning, but in this sentence it is functioning as the predicate.

It comes from the root related to opening and literally means opened/open.

You can think of it in two ways:

• as an adjective meaning open
• as the predicate in the office is open

Both ideas are helpful, but grammatically in this sentence its role is the khabar (predicate).

Yes. When a predicate is adjective-like, it normally agrees with the noun in gender and number.

Here:

• المكتب is singular masculine
• so مفتوح is also singular masculine

For example:

• النافذة مفتوحة = the window is open
  because النافذة is feminine singular

• الأبواب مفتوحة = the doors are open
  because الأبواب is plural

طيلة means throughout, for the whole of, or over the entire duration of.

So:

• طيلة اليوم = throughout the day / all day long

It is used to express duration across the full span of something.

You may also see similar expressions such as:

• طوال اليوم = all day / throughout the day
• طيلة الأسبوع = throughout the week
• طيلة السنة = throughout the year

Because اليوم by itself usually just means today or the day.
Adding طيلة gives the idea of for the entire duration of the day.

Compare:

• اليوم = today
• طيلة اليوم = all day / throughout the day

So طيلة is what adds the sense of continuous duration.

Because طيلة is followed by the time period that it covers.

So the structure is:

• طيلة + time noun

Examples:

• طيلة اليوم = throughout the day
• طيلة الليل = throughout the night
• طيلة الأسبوع = throughout the week

This works a bit like a possessed construction (iḍāfa), where طيلة is connected to the word after it.

In fully vocalized formal Arabic, it would typically be:

المكتبُ مفتوحٌ طيلةَ اليومِ

Breakdown:

• المكتبُ — nominative, because it is the subject (mubtadaʾ)
• مفتوحٌ — nominative, because it is the predicate (khabar)
• طيلةَ — accusative, because it functions as an adverbial expression of time/duration
• اليومِ — genitive, because it comes after طيلة in this construction

In normal everyday writing, these endings are usually omitted.

A careful formal pronunciation is:

al-maktabu maftūḥun ṭīlata al-yawmi

If you pause at the end, the final case ending is often not pronounced fully, and natural pronunciation may sound closer to:

al-maktab maftūḥ ṭīlat al-yawm

For a learner, a good practical pronunciation is:

al-maktab maftūḥ ṭīlat al-yawm

Yes. طوال اليوم is very common and means almost the same thing: all day / throughout the day.

So both are natural:

• المكتب مفتوح طيلة اليوم
• المكتب مفتوح طوال اليوم

They are very close in meaning. A learner will hear both.