Noudat ons goedkeuring gekry het, mag die aanbouing begin.

Noudat means now that. It introduces the reason or background for what follows.

So in this sentence, Noudat ons goedkeuring gekry het sets up the idea now that we have received approval.

In standard Afrikaans, noudat is a fixed conjunction meaning now that, so it is normally written as one word.

It works like a single linking word that introduces a subordinate clause.

Because noudat introduces a subordinate clause.

In a normal main clause, you would say:

• Ons het goedkeuring gekry.

But after a subordinating conjunction like noudat, Afrikaans pushes the finite verb to the end. In the perfect tense, that gives:

• ... ons goedkeuring gekry het

So:

• main clause: ons het gekry
• subordinate clause: ... gekry het

Because the sentence starts with the subordinate clause:

• Noudat ons goedkeuring gekry het, ...

That whole clause takes the first position in the sentence. In Afrikaans main clauses, the finite verb then comes next. So the order becomes:

• Noudat ... , mag die aanbouing begin.

This is the same kind of inversion you see in other Afrikaans sentences when something other than the subject comes first.

Here mag means may / is allowed to.

That fits the context, because the sentence talks about having received approval. The idea is not mainly ability, but permission.

• mag = may, be allowed to
• kan = can, be able to

So mag is the natural choice here.

After a modal verb like mag, Afrikaans uses the bare infinitive.

So you say:

• mag begin
• kan begin
• moet begin

Not:

• mag te begin

The word te is used in other infinitive structures, but not after modal verbs.

Aanbouing usually means an extension, addition, or part added onto a building.

Depending on context, it can refer to:

• the physical extension itself, or
• the building project/work on that extension

In this sentence, it most naturally means the extension project or the extension work that can now start.

Die is the definite article in Afrikaans, meaning the.

Afrikaans uses die for definite nouns regardless of gender, and it is also used for plural nouns. Here it simply means:

• die aanbouing = the extension

It suggests a specific extension already known from the context.

It combines two different time ideas:

• ons goedkeuring gekry het = perfect tense
  This shows the approval has already been received.
• mag ... begin = present-form modal construction
  This expresses current permission, often with a future sense: the extension may now begin.

So the overall meaning is: the approval happened first, and as a result the building work is now allowed to start.

Yes, that is also grammatical.

The difference is mostly one of focus:

• Noudat ons goedkeuring gekry het, mag die aanbouing begin.
  This emphasizes the reason/background first.
• Die aanbouing mag begin noudat ons goedkeuring gekry het.
  This emphasizes the main event first.

Both are natural; the original version just foregrounds the approval as the key setup.