بعد ما اشتريت بطاطا من السوق، رجعت عالبيت.

In Levantine, بعد ما + verb is a very common way to say after doing something or after something happened.

So:

• بعد ما اشتريت... = after I bought...

By itself, بعد can also mean things like after, later, or even still/yet depending on context. Adding ما before a full verb clause makes the after meaning very clear.

Because the verb already tells you the subject.

In اشتريت, the ending -ت shows I in the past tense, so اشتريت already means I bought.

That is why Arabic often leaves out subject pronouns when they are not needed.
You could say أنا اشتريت if you want emphasis, contrast, or extra clarity, but normally it is not necessary.

اشتريت is the past tense, first person singular form of the verb اشترى / يشتري, meaning to buy.

So:

• اشتريت = I bought
• اشتريتِ = you bought (to a woman, depending on pronunciation/context)
• اشترى = he bought
• اشترت = she bought

In this sentence, context makes it clear that اشتريت means I bought.

Here it most naturally means potatoes in a general or collective sense.

In Levantine, بطاطا often works like a mass/collective noun when talking about the food item in general, so it does not have to match English singular/plural in a neat one-to-one way.

If you specifically want to say one potato, you would often say:

• حبة بطاطا = one potato

So in this sentence, اشتريت بطاطا is best understood as I bought potatoes.

Because with buy, Arabic often uses من to show the source of what you bought.

So:

• اشتريت بطاطا من السوق = I bought potatoes from the market

This is very natural Arabic.

If you said في السوق or بالسوق, that would focus more on the location where the buying happened:

• اشتريت بطاطا بالسوق = I bought potatoes at the market

Both ideas are possible, but من السوق is extremely normal when talking about where something was bought from.

All of those are possible translations depending on the context.

رجعت comes from رجع, which means to return / go back / come back.

So:

• رجعت عالبيت can mean
  
  • I went back home
  • I came back home
  • I returned home

English chooses among those based on perspective, but the Arabic verb itself covers that general idea of returning.

عالبيت is a colloquial spelling of على البيت.

In Levantine, على is very often shortened in speech, and when it comes before الـ, it is commonly written as عالـ.

So:

• على البيت → عالبيت

This is very normal in informal Levantine writing.

Meaning:

• عالبيت = to the house / home

Because prepositions do not always match English word-for-word.

Yes, على often means on, but in dialect it is also used in expressions of movement and direction where English would use to.

So رجعت عالبيت does not mean I returned on the house.
It simply means I went back home.

In Levantine, this is a very natural way to say it. You may also hear:

• رجعت للبيت

That also means I went back home / returned to the house.

Yes, absolutely.

Both are natural:

• بعد ما اشتريت بطاطا من السوق، رجعت عالبيت
• رجعت عالبيت بعد ما اشتريت بطاطا من السوق

The version in your sentence puts the after-clause first, which helps set up the sequence of events.
The second version starts with the main action and then adds when it happened.

So the difference is mostly about focus and flow, not basic meaning.

Yes, it has a clear Levantine feel, especially عالبيت.

A more formal / MSA-style version might be:

• بعدما اشتريتُ بطاطا من السوق، رجعتُ إلى البيت
• or more formally, بعدما اشتريتُ بطاطا من السوق، عدتُ إلى البيت

What makes your original sentence feel colloquial/Levantine:

• عالبيت instead of إلى البيت
• no case endings
• overall spoken-dialect structure

So your sentence is very natural for everyday Levantine Arabic.