Bij het station werkt het poortje niet, dus de medewerker laat ons zonder kaartje door.

Dutch main clauses follow the verb-second rule: the finite verb usually comes in the second position, but the first position can be something other than the subject.

So in:

Bij het station werkt het poortje niet

the first position is Bij het station, and the finite verb werkt comes second. The subject het poortje comes after the verb.

A more basic order would be:

Het poortje werkt niet bij het station

but that sounds less natural here. Starting with Bij het station sets the scene first.

Bij often means at or near, while in means in / inside.

So:

• bij het station = at the station
• in het station = inside the station building

In this sentence, bij het station is a natural way to refer to the station area, especially where the gate is.

Poortje is the diminutive form of poort.

• poort = gate
• poortje = little gate / gate / ticket gate

In Dutch, diminutives are used very often, not only to mean small, but also for everyday objects in a natural, idiomatic way. For ticket barriers at stations, poortje is very common.

Also, all diminutives are het-words, which is why it is het poortje.

Because both nouns are het-words:

• het station
• het poortje

With station, you just have to learn the gender with the noun.

With poortje, it is easier: any noun ending in a diminutive such as -je, -tje, -pje, -kje is always a het-word.

Because the subject is singular:

het poortje

So the verb must also be singular:

• het poortje werkt
• de poortjes werken

Even though Bij het station comes first, it is not the subject. The subject is still het poortje.

In a simple main clause, niet usually comes after the verb and after the elements that belong closely with it, but before whatever comes later in the sentence.

Here:

Bij het station werkt het poortje niet

This is the normal way to say that the gate is not working.

You can think of niet werken as a common combination meaning to not function.

Dus means so.

It is a coordinating conjunction, so it does not force inversion. That means the next clause keeps normal main-clause word order:

dus de medewerker laat ons zonder kaartje door

Subject first, then verb:

• de medewerker = subject
• laat = finite verb

This is different from words like daarom, which do trigger inversion:

• Dus de medewerker laat ons door.
• Daarom laat de medewerker ons door.

That difference is very important in Dutch.

Because ons is the object form, while wij is the subject form.

Here, de medewerker is the subject, and ons is the direct object:

• de medewerker lets
• ons through

Compare:

• Wij gaan door. = We go through.
• De medewerker laat ons door. = The employee lets us through.

So wij would be wrong here.

Because the verb here is the separable verb doorlaten:

• doorlaten = to let through

In a main clause, separable verbs split:

• De medewerker laat ons door.

So:

• laat = the conjugated part
• door = the separable prefix, which goes to the end

This is very common in Dutch:

• opbellen → Ik bel je op.
• meenemen → Zij neemt het mee.
• doorlaten → Hij laat ons door.

After zonder (without), Dutch usually uses the noun without an article when speaking in a general sense.

So:

• zonder kaartje = without a ticket
• zonder jas = without a coat
• zonder probleem = without a problem

Using zonder een kaartje is possible in some contexts, but zonder kaartje is the normal, idiomatic choice here.

In everyday Dutch, kaartje is the normal word for a ticket.

• kaart often means card, map, or sometimes ticket depending on context
• kaartje very often means a small ticket, admission ticket, travel ticket, etc.

So at a station, kaartje sounds very natural.

Like poortje, this is also a diminutive, but in many cases Dutch uses the diminutive simply because that is the standard word people say.

Yes, but the meaning or emphasis would change slightly.

• de medewerker = the employee / staff member
  This sounds like a specific person involved in the situation.
• een medewerker = an employee / a staff member
  This introduces the person less specifically.
• het personeel = the staff
  This refers more generally to the employees as a group.

So de medewerker suggests a particular employee is helping them through the gate.

Yes. Dutch often uses the simple present for current situations:

• werkt niet = is not working
• laat ons door = lets us through / is letting us through

This is very normal. Dutch uses the simple present more often than English does, where English may prefer a progressive form such as is not working or is letting us through.