مكتب البريد قريب من المصعد في هذا الطابق.

Because this is a present-tense nominal sentence in Arabic. In Modern Standard Arabic, you usually do not say a separate word for is / are in simple present sentences.

So:

• مكتب البريد قريب من المصعد
  literally: The post office near the elevator
• natural English: The post office is near the elevator

If you wanted was, Arabic would use a verb such as كان.

This is because مكتب البريد is an iḍāfa construction, often called a genitive construction or noun-noun phrase.

It literally means something like:

• مكتب = office
• البريد = the mail / the post

Together:

• مكتب البريد = the post office
  literally: office of the mail

In an iḍāfa:

• the first noun usually does not take ال
• the second noun can make the whole phrase definite

So even though مكتب has no ال, the whole phrase is definite because البريد is definite.

Because it agrees with مكتب, which is the head noun of مكتب البريد.

• مكتب = masculine singular
• so the predicate adjective is قريب = masculine singular

Even though البريد comes right after it, the phrase is still centered on مكتب.

If the subject were feminine, you would usually use قريبة instead.

Grammatically, قريب is an adjective, but here it is being used as the predicate of the sentence.

So:

• مكتب البريد = the subject
• قريب من المصعد = the predicate, meaning is near the elevator

Arabic often uses adjectives this way in nominal sentences.

Because the normal pattern for near when followed by a place or thing is:

• قريب من = near / close to

So:

• قريب من المصعد = near the elevator

The preposition من is the usual one after قريب in this meaning.

Because المصعد means the elevator. In this sentence, the elevator is treated as a specific, identifiable one.

Arabic often uses the definite article when the speaker assumes the listener can identify the thing being mentioned.

Also, after a preposition like من, the noun can still be definite:

• من المصعد = from / near the elevator

The ال does not disappear after a preposition.

Because طابق is a masculine noun, so it takes the masculine demonstrative:

• هذا = this (masculine)
• هذه = this (feminine)

So:

• هذا الطابق = this floor

If the noun were feminine, you would use هذه instead.

In Modern Standard Arabic, when a demonstrative like هذا comes before a noun, that noun is usually also definite with ال.

So the normal pattern is:

• هذا + الـnoun
• هذه + الـnoun

Therefore:

• هذا الطابق = this floor

This is different from English, where we say this floor, not this the floor.

Yes. Arabic word order is somewhat flexible.

The given sentence:

• مكتب البريد قريب من المصعد في هذا الطابق

is natural and neutral.

But you could also say:

• في هذا الطابق، مكتب البريد قريب من المصعد

This gives a little more emphasis to on this floor.

So the original order is normal, but other orders are possible for style or emphasis.

In normal context, it most naturally means the post office is on this floor, and it is near the elevator.

So the overall idea is:

• On this floor, the post office is near the elevator.

In real usage, context usually makes this clear. The phrase at the end often applies to the whole situation rather than only to the nearest noun.

With full vowel endings, it would be:

مَكْتَبُ البَرِيدِ قَرِيبٌ مِنَ المِصْعَدِ فِي هَذَا الطَّابِقِ

Here is why:

• مكتبُ: nominative, because it is the subject
• البريدِ: genitive, because it is the second term of the iḍāfa
• قريبٌ: nominative, because it is the predicate
• المصعدِ: genitive, because it comes after من
• الطابقِ: genitive, because it comes after في
• هذا does not visibly change form here

In normal unvowelled writing, these endings are usually not written.

A helpful pronunciation is:

maktabu l-barīdi qarībun mina l-miṣʿadi fī hādhā ṭ-ṭābiqi

A few points:

• ق is a deep q sound, not an English k
• ص and ط are emphatic consonants
• ع in المصعد is a throat sound that English does not have
• الطابق is pronounced with assimilation of ال to ط, so it sounds like aṭ-ṭābiq

You do not need perfect pronunciation immediately, but those are the main things to notice.

Because ط is a sun letter. When ال comes before a sun letter, the l sound is not pronounced, and the next consonant is doubled.

So:

• written: الطابق
• pronounced: aṭ-ṭābiq

This is a standard pronunciation rule in Arabic spelling and reading.

Yes, but it would slightly change the structure.

The original sentence is a straightforward statement:

• مكتب البريد قريب من المصعد في هذا الطابق
• The post office is near the elevator on this floor

If you used يوجد, you would be saying something more like:

• يوجد مكتب بريد قريب من المصعد في هذا الطابق
• There is a post office near the elevator on this floor

So both are possible, but the original sentence is not an existential there is sentence; it is a normal nominal sentence.

Because in an iḍāfa construction, the first noun normally does not take ال.

So:

• correct: مكتب البريد
• not standard here: المكتب البريد

If you put ال on the first noun, it is no longer the normal iḍāfa pattern.

That is one of the key things learners have to get used to: in Arabic, a phrase can be definite even if the first noun has no ال.