Ved ankomst til byen ringer hun til mig med det samme.

Here ved means something like upon, on, or at the time of. So ved ankomst is a set-like expression meaning upon arrival.

It is built from:

• ved = at/upon
• ankomst = arrival

So Ved ankomst til byen ... means Upon arrival in/to the city ...

Danish often uses a bare noun in fixed expressions like this, especially after certain prepositions.

So:

• ved ankomst = upon arrival

This sounds natural and general.
Ved ankomsten is possible, but it sounds more specific, like upon the arrival or upon her/the arrival, with stronger reference to one particular arrival.

In many formal or written expressions, Danish prefers the shorter form:

• ved ankomst
• ved afrejse
• ved betaling

Because Danish main clauses normally follow the V2 rule: the finite verb comes in the second position.

In this sentence, Ved ankomst til byen is placed first. That whole phrase counts as the first element, so the verb ringer must come next, before the subject hun.

Compare:

• Hun ringer til mig med det samme.
• Ved ankomst til byen ringer hun til mig med det samme.

So the word order changes because the sentence begins with a time expression.

In Danish, the verb at ringe usually takes the preposition til when it means to call/phone someone.

So you say:

• Hun ringer til mig. = She calls me.

This is different from English, where call someone does not need a preposition.

So for English speakers, this is an important pattern to remember:

• at ringe til nogen = to call someone

Med det samme is a common fixed expression meaning:

• immediately
• right away
• at once

Literally it looks like with the same, but that literal meaning is not what matters here. You should learn it as a whole expression.

Very close alternatives are:

• straks
• øjeblikkeligt (more formal)

So:

• Hun ringer til mig med det samme.
• Hun ringer til mig straks.

Both are fine, but med det samme is very common in everyday Danish.

Because til shows movement toward a destination, while i usually shows location inside a place.

Here the noun ankomst implies movement to the city, so til byen is natural:

• ankomst til byen = arrival in/to the city

Compare:

• Hun kommer til byen. = She comes to the city.
• Hun er i byen. = She is in the city / in town.

So til fits the idea of arriving somewhere.

Byen means the city/town. The definite form is used because the place is understood to be specific or known from context.

So the speaker and listener both know which city is meant, or it has already been mentioned.

Compare:

• til byen = to the city / to town
• til en by = to a city (any city, not specific)

In real usage, Danish often uses the definite form when the place is already identifiable from the situation.

Yes. Ankomst is the noun arrival, and ankomme is the verb to arrive.

So these are related:

• at ankomme = to arrive
• ankomst = arrival

That means the sentence could also be expressed with a verb clause, for example:

• Når hun ankommer til byen, ringer hun til mig med det samme.

That version is a little more explicit, while ved ankomst til byen is more compact and slightly more formal.

Yes, absolutely. That is a very natural alternative.

The difference is mainly in style:

• Ved ankomst til byen ringer hun til mig med det samme.

  • more compact
  • slightly more formal or written

• Når hun ankommer til byen, ringer hun til mig med det samme.

  • more explicit
  • often feels more conversational and easier for learners to understand

Both are correct.

Normally, no.

Ved ankomst til byen is just a fronted prepositional phrase, not a subordinate clause, so you do not put a comma after it in standard Danish punctuation.

So the sentence is correctly written as:

• Ved ankomst til byen ringer hun til mig med det samme.

But if you use a subordinate clause instead, then you do use a comma:

• Når hun ankommer til byen, ringer hun til mig med det samme.

So the comma depends on the structure, not just on the pause in speech.