الاسبوع الماضي كان عندي امتحان.

A common Levantine pronunciation would be:

l-isbuuʕ il-maaDi kaan ʕandi imtiHaan

A few notes:

• الاسبوع is often pronounced l-isbuuʕ or l-usbuuʕ, depending on the speaker.
• الماضي is usually il-maaDi in Levantine.
• عندي starts with the sound ʕ (the letter ع), which does not exist in English.
• امتحان is usually imtiHaan.

So the rhythm is roughly:

l-isbuuʕ il-maaDi | kaan ʕandi imtiHaan

Because the two words do different jobs.

• الماضي tells you when: last week
• كان puts the situation itself in the past

In Levantine, عندي by itself means I have or literally at me / with me.
To say I had, you usually add كان:

• عندي امتحان = I have an exam
• كان عندي امتحان = I had an exam

So even though الاسبوع الماضي already gives a past time, كان is still the normal way to make عندي past.

عندي literally means at me.

It is made of:

• عند = at / by / with
• -ي = my / me

So:

• عندي = at me
• in natural English, this often becomes I have

This is very common in Arabic. Instead of using one basic verb like English to have, Arabic often expresses possession with a structure like at me, with me, or for me.

So:

• عندي كتاب = I have a book
• كان عندي امتحان = I had an exam

Because in Arabic, adjectives normally come after the noun.

So:

• الاسبوع = the week
• الماضي = past / last

Together:

• الاسبوع الماضي = literally the week the past
• natural English: last week

This is the normal Arabic word order:

• noun first
• adjective second

The same pattern appears in many other phrases:

• البيت الكبير = the big house
• الشهر الماضي = last month
• السنة الماضية = last year

Because Arabic adjectives usually match the noun in definiteness.

Here, الاسبوع is definite because it has الـ.
So the adjective describing it, الماضي, also becomes definite.

That is why you say:

• الاسبوع الماضي

not:

• الاسبوع ماضي

This agreement is very normal in Arabic:

• الطالب الجديد = the new student
• البنت الطويلة = the tall girl
• الاسبوع الماضي = last week

Because the meaning is an exam, not the exam.

So:

• امتحان = an exam / exam
• الامتحان = the exam

In the sentence كان عندي امتحان, the speaker is mentioning an exam as one event, not necessarily a specific already-known exam.

If the exam were already specific in the conversation, الامتحان could be possible:

• كان عندي الامتحان مبارح = I had the exam yesterday

But in your sentence, امتحان is the natural choice for an exam.

Usually, no.

Arabic often uses time expressions without a preposition:

• اليوم = today
• امبارح = yesterday
• الاسبوع الماضي = last week

So الاسبوع الماضي كان عندي امتحان is completely normal.

English often says last week without in too, so in this sentence the two languages happen to work similarly.

In some contexts, speakers may use a preposition like بـ or في, but here it is not needed.

Yes, that is also natural.

Both of these work:

• الاسبوع الماضي كان عندي امتحان
• كان عندي امتحان الاسبوع الماضي

The difference is mainly emphasis:

• الاسبوع الماضي كان عندي امتحان puts the time first, like Last week, I had an exam
• كان عندي امتحان الاسبوع الماضي starts with the main event, like I had an exam last week

Arabic is often flexible with word order, especially with time expressions.

It is very natural in Levantine, and it is also easy to understand for speakers of Standard Arabic.

A more formal Standard Arabic spelling would usually be:

الأسبوع الماضي كان عندي امتحان

The main differences are:

• Standard Arabic usually writes the hamza in الأسبوع
• Levantine writing is often more relaxed, so people may write الاسبوع
• In actual Levantine speech, there are no case endings

Also, كان عندي is very common in both everyday dialect and widely understandable Arabic.

Because الأسبوع is the more standard spelling, with the hamza written.

In casual dialect writing, many people leave hamzas out, so you may see:

• الأسبوع
• الاسبوع

Both refer to the same word.

So if you are learning carefully or writing more formally, use:

الأسبوع الماضي

If you are reading texts, messages, or social media in dialect, do not be surprised to see:

الاسبوع الماضي

Yes. امتحان is very common for exam.

Another possible word is:

• اختبار

In many contexts, both can mean test/exam, though امتحان is extremely common in everyday speech for school or university exams.

So these are both possible:

• كان عندي امتحان
• كان عندي اختبار

For many learners, امتحان is the more useful everyday word to recognize first.