يعمل صديقي في الإدارة.

A common pronunciation is yaʿmalu ṣadīqī fī al-idārah.

If you want the fully vocalized form, it is:

يَعْمَلُ صَدِيقِي فِي الْإِدَارَةِ

A few quick notes:

• يعمل = yaʿmalu
• صديقي = ṣadīqī
• في = fī
• الإدارة = al-idārah

In careful MSA, the last vowel in يعمل is pronounced, but in pause people often shorten things slightly in natural reading.

Because verb-first word order is very common in Modern Standard Arabic.

So:

يعمل صديقي في الإدارة
literally follows the pattern works my friend in administration.

This is completely normal in Arabic.
You can also say:

صديقي يعمل في الإدارة

That is also correct, but it puts more attention on my friend as the topic.

يعمل means he works or he is working, depending on context.

It comes from the root ع-م-ل, which is connected with work / doing / labor.

In this sentence, يعمل is:

• 3rd person
• masculine
• singular
• present/imperfect tense

So it matches صديقي = my friend.

Because Arabic verbs already show the subject information.

In يعمل, the form itself already tells you:

• he
• singular
• masculine

So Arabic does not need a separate pronoun here.

That means:

• يعمل = he works not just works

Then صديقي is added as the explicit subject: my friend.

The basic noun is صديق = friend.

To say my friend, Arabic adds the possessive suffix ـي, which means my:

• صديق = friend
• صديقي = my friend

So ـي is a very common ending meaning my.

For example:

• كتابي = my book
• بيتي = my house
• صديقي = my friend

Because in a verb-first sentence, the subject often comes after the verb.

So the structure here is:

• يعمل = verb
• صديقي = subject
• في الإدارة = prepositional phrase

This is one of the standard sentence patterns in MSA.

في means in or sometimes at, depending on context.

الإدارة means administration, management, or the administrative department.

So في الإدارة can mean:

• in administration
• in the administration department
• in management

The best English translation depends on context.

الـ is the Arabic definite article, meaning the.

So:

• إدارة = administration / management
• الإدارة = the administration / the management / the administrative department

Arabic often uses the definite article in places where English may or may not use the.
So even if English says in administration, Arabic may naturally say في الإدارة.

Yes. That sentence is also correct:

صديقي يعمل في الإدارة

Both are grammatical, but the emphasis is a little different:

• يعمل صديقي في الإدارة = more neutral, often more formal/written
• صديقي يعمل في الإدارة = starts with my friend, so it highlights the subject more

Both are good MSA.

In MSA, when the verb comes before the subject, the verb is typically singular.

So with a singular subject like صديقي, that is exactly what you expect:

• يعمل صديقي = my friend works

This is normal Arabic grammar.

You would change both the noun and the verb:

تعمل صديقتي في الإدارة.

That means My female friend works in administration.

Changes:

• صديقي = my male friend
• صديقتي = my female friend
• يعمل = he works
• تعمل = she works

In normal Arabic writing, short vowels are usually not written.

So you see:

يعمل صديقي في الإدارة

But the fully vocalized version is:

يَعْمَلُ صَدِيقِي فِي الْإِدَارَةِ

Learners should know that:

• regular printed Arabic usually leaves these marks out
• dictionaries, textbooks, and beginner materials may include them
• you learn to supply them from grammar and context

So the sentence you were given is the normal everyday written form.