بعد العمل أذهب إلى مكتب البريد لأن عندي طردا لصديقتي.

No preposition is missing. بعد itself means after and is commonly used as a time expression.

So بعد العمل means after work.

If you add full case endings, it would be:

بعدَ العملِ

• بعدَ = after
• العملِ = the work / work, in the genitive because it follows بعد

This is a very common pattern in Arabic:

• بعد الدرس = after the lesson
• بعد الغداء = after lunch

In Arabic, the present tense can express:

• a habitual action: I go
• a near-future or planned action: I am going / I will go

So بعد العمل أذهب can mean something like After work, I go or After work, I’m going depending on context.

If you want to make the future more explicit, you could say:

بعد العمل سأذهب إلى مكتب البريد

which clearly means After work, I will go to the post office.

The verb ذهب normally uses إلى to show movement toward a place.

So:

• أذهب إلى البيت = I go to the house
• أذهب إلى المدرسة = I go to school
• أذهب إلى مكتب البريد = I go to the post office

This is just the normal pattern: ذهب إلى = to go to.

This is an iḍāfa construction, often called a construct phrase.

Literally, it is:

• مكتب = office
• البريد = the mail / post

So مكتب البريد literally means the office of the mail, which is the natural Arabic way to say post office.

A useful point about iḍāfa:

• the first noun usually does not take ال
• the whole phrase becomes definite because the second noun is definite

With full case endings after إلى, it would be:

إلى مكتبِ البريدِ

لأن means because.

In fully vocalized Arabic, it is usually written لأنَّ.

It introduces the reason:

• أذهب إلى مكتب البريد = I go to the post office
• لأن عندي طردًا لصديقتي = because I have a package for my friend

So the whole sentence gives an action and then the reason for it.

Arabic often expresses possession with a phrase like عندي, which literally means at me or with me.

So:

• عندي كتاب = I have a book
• عندي طرد = I have a package

This is much more common in everyday Arabic than using a verb meaning to possess.

So in your sentence:

لأن عندي طردًا لصديقتي

literally feels like:

because at me there is a package for my friend

but in natural English we simply say:

because I have a package for my friend

This is a very good grammar question.

Here the word is really طردًا in fully vocalized form, and that accusative ending is there because of لأنَّ.

Without لأنَّ, you would normally say:

عندي طردٌ

Here طردٌ is nominative.

But after لأنَّ, the noun connected to it becomes accusative. In this sentence, عندي comes first, and طردًا is delayed, but it is still the noun affected by لأنَّ.

So:

• عندي طردٌ = I have a package
• لأن عندي طردًا = because I have a package

That is why طردًا is correct here.

لصديقتي breaks down like this:

• لِـ = for / to
• صديقة = female friend
• ـي = my

So لصديقتي means for my female friend or sometimes to my female friend, depending on context.

In this sentence, for my female friend is the most natural meaning:

• طردًا لصديقتي = a package for my female friend

If the friend were male, it would be:

لصديقي

Not necessarily.

In Modern Standard Arabic, صديقة basically means female friend, and صديقتي means my female friend.

In some contexts, it could refer to a romantic partner, but the word by itself does not force that meaning. Context decides.

So in a basic learning sentence like this, the safest understanding is:

my female friend

Yes. You could say:

لأن لديَّ طردًا لصديقتي

This also means because I have a package for my friend.

Very roughly:

• عندي = very common, natural
• لديّ = also common, often a bit more formal or written

Both are correct in Modern Standard Arabic, and learners should recognize both.