El coche tardó en arrancar, pero al final arrancó después de unos minutos.

Because tardar normally takes en + infinitive when it means to take time to do something.

• El coche tardó en arrancar = The car took a while to start.
• Tardó en llegar = He/She took a while to arrive.
• Tardamos en terminar = We took a while to finish.

So en is part of the standard pattern here.

Here arrancar means to start, specifically about a car engine.

In other contexts, arrancar can also mean things like:

• to pull out / uproot
  
  • arrancar una planta
• to begin suddenly
• to drive off / pull away

But with el coche, the most natural meaning is to start.

The two instances do different jobs:

• tardó en arrancar = it took a while to start
• al final arrancó = in the end, it did start

So the first mentions the delayed action as an infinitive, and the second confirms that the action finally happened as a completed event.

English does something similar:

• The car took a while to start, but in the end it started.

Because the sentence describes completed events in the past, so Spanish uses the preterite:

• tardó
• arrancó

The speaker is treating the whole situation as a finished episode: the car had trouble starting, and then it started.

If you used the imperfect, it would sound more like background, habit, or an unfinished situation:

• El coche tardaba en arrancar could suggest a repeated or descriptive situation, like The car used to take a while to start.

Al final means in the end or eventually.

In this sentence:

• pero al final arrancó = but in the end it started

A useful distinction:

• al final often focuses on the final outcome of a situation
• finalmente can also mean finally / finally at last, and is sometimes a bit more formal

In everyday speech, al final is very common.

Because después normally needs de before a noun phrase.

• después de unos minutos = after a few minutes
• después de la cena = after dinner
• después de eso = after that

But when después stands alone, it can mean later / afterwards:

• Arrancó y después nos fuimos = It started and afterwards we left.

So:

• después de + noun/pronoun
• después alone = afterwards/later

Here unos minutos means a few minutes or some minutes.

Unos/unas can mean:

1. some
2. approximately / a few

In this sentence, it gives an approximate amount of time, not an exact count.

• después de unos minutos = after a few minutes

It sounds natural and slightly imprecise, which is very common in everyday Spanish.

Here el is simply the normal definite article: the car.

Spanish uses articles very often, and in this sentence it is completely expected because the speaker is talking about a specific car in a specific situation.

So:

• El coche tardó en arrancar = The car took a while to start

There is nothing unusual here; Spanish just normally includes the article before a singular countable noun like coche.

Sometimes, but not with exactly the same meaning.

For cars:

• arrancar = to start the engine / for the car to start
• encender = to switch on / turn on

You can say:

• No consigo arrancar el coche = I can’t get the car to start.
• Encendí las luces = I turned on the lights.

With a car, arrancar is the more precise verb for the engine starting. Encender el coche may be understood in some contexts, but arrancar is the normal choice here.

Because el coche is the subject of both verbs:

• the car took a while
• the car started

Spanish often allows things to be phrased this way with inanimate subjects.

The structure is:

• [subject] + tardar en + infinitive

Examples:

• El ordenador tardó en encenderse = The computer took a while to turn on.
• La puerta tardó en abrirse = The door took a while to open.

So no indirect object pronoun is needed here.

Yes. In Spain, coche is the most common everyday word for car.

In much of Latin America, people often say:

• carro
• auto
• automóvil in more formal contexts

So this sentence sounds very natural for Spain Spanish.

Yes, and this sentence shows the intransitive use.

• El coche arrancó = The car started.
  Here the car itself is the subject.

You can also use it transitively in some contexts:

• Arranqué el coche = I started the car.

So arrancar can mean either:

• to start
• to start something

depending on the structure.

Yes, absolutely. That is also very natural.

• El coche tardó en arrancar = The car took a while to start.
• El coche tardó unos minutos en arrancar = The car took a few minutes to start.

The version in your sentence splits the idea into two parts:

1. it took a while to start
2. it finally started after a few minutes

That makes the sequence a bit more explicit.