Metroens indgang er tæt på stationen, men udgangen ligger på den anden side af bygningen.

Metroens is the genitive form, meaning the metro’s.

It is built like this:

• metroen = the metro
• metroens = the metro’s

So Danish often adds -s to show possession, much like English apostrophe-s, but normally without an apostrophe:

• mandens bil = the man’s car
• skolens navn = the school’s name

So Metroens indgang means the metro’s entrance.

This is a very common Danish pattern.

After a possessive or genitive word like metroens, the following noun is usually indefinite in form:

• metroens indgang = the metro’s entrance
• not metroens indgangen

That is because the possessive already makes it definite enough.

But in the second clause, udgangen stands on its own, so Danish marks it as definite directly on the noun:

• udgangen = the exit

So the difference is:

• metroens indgang = possessive structure
• udgangen = definite noun by itself

Yes. They are a very useful pair to learn.

• indgang = entrance
• udgang = exit

They are formed from:

• ind- = in
• ud- = out
• gang = historically passage / going / way

So they are closely related and easy to remember as opposites:

• indgang = way in
• udgang = way out

Tæt på means close to / near.

It is a fixed expression:

• tæt = close / tight
• på = the preposition used here

So:

• tæt på stationen = close to the station

English speakers sometimes want to translate word by word and expect a different preposition, but in Danish tæt på is the normal expression.

Other examples:

• Jeg bor tæt på skolen. = I live close to the school.
• Butikken ligger tæt på her. = The shop is close by.

Because Danish adds the definite article to the end of the noun.

• station = a station
• stationen = the station

So tæt på stationen means close to the station, not just close to a station.

This is one of the biggest differences from English: Danish often puts the at the end of the noun instead of using a separate word.

Because Danish normally uses double definiteness when a definite singular noun has an adjective in front of it.

So you get:

• den
  • adjective + noun in definite form

Here:

• den = definite article
• anden = other / second
• side becomes the definite idea side(n), but in this fixed expression you say den anden side

So den anden side means the other side.

This pattern is very common:

• den store bil = the big car
• det gamle hus = the old house
• den anden side = the other side

It literally means on the other side of the building.

The structure is:

• på = on
• den anden side = the other side
• af bygningen = of the building

In Danish, på den anden side af ... is the standard way to say on the other side of ....

Also:

• bygning = building
• bygningen = the building

So af bygningen means of the building or more naturally in English, of the building / around the building depending on context.

Because Danish often uses ligge for the physical location of places or things.

• ligger = lies / is situated
• er = is

So:

• udgangen ligger på den anden side af bygningen = the exit is located on the other side of the building

Using ligger sounds natural when talking about where something is situated geographically or physically.

Compare:

• Skolen ligger ved siden af kirken. = The school is next to the church.
• Nøglen er på bordet. = The key is on the table.

Both can describe location, but ligger is especially common for places, buildings, entrances, exits, shops, towns, etc.

Because men is a coordinating conjunction, and after it Danish keeps normal main-clause word order:

• subject + verb

So:

• men udgangen ligger ...

This is like English:

• but the exit is ...

Inversion does not happen just because of men.

You often get inversion in Danish after certain sentence adverbs or fronted elements, for example:

• På den anden side ligger udgangen.

But after men, the usual order stays:

• men udgangen ligger ...

Yes — this sentence is full of it, and that makes it a great example.

You can see several different ways Danish shows definiteness:

• metroen = the metro
• metroens = the metro’s
• stationen = the station
• udgangen = the exit
• den anden side = the other side
• bygningen = the building

So one useful lesson from this sentence is that Danish does not use just one single word for the the way English does. It can show definiteness by:

• a suffix on the noun: -en
• a genitive form: metroens
• a separate article plus adjective: den anden

That mix is very normal in Danish.