Čím dřív nastoupíme do tramvaje, tím dřív budeme v centru.

This is a very common Czech pattern meaning the ... , the ....

So:

• Čím dřív ...
• tím dřív ...

means:

• The sooner ...
• the sooner ...

It works like a comparison of degree or extent:

• Čím víc studuješ, tím víc umíš. = The more you study, the more you know.
• Čím dřív vyrazíme, tím líp. = The sooner we leave, the better.

In your sentence:

• Čím dřív nastoupíme do tramvaje, tím dřív budeme v centru.
• The sooner we get on the tram, the sooner we’ll be in the center.

Here čím is not the ordinary question word what? It is the instrumental form of co used in this fixed comparative structure:

• čím ... tím ...

You do not translate it literally as by what here. You should learn the whole pattern as one unit:

• čím větší, tím lepší = the bigger, the better
• čím dřív, tím líp = the sooner, the better

So in this construction, čím is simply the correct form.

Dřív means earlier / sooner.

It is the comparative form related to brzy or brzo (early / soon).

So:

• brzy / brzo = early, soon
• dřív / dříve = earlier, sooner

In this sentence, dřív fits naturally because the whole idea is comparative:

• the sooner we get on
• the sooner we’ll be in the center

Yes, but only a small one.

• dřív is very common in everyday spoken and written Czech.
• dříve is a bit more formal or stylistically neutral/bookish.

In many sentences they are interchangeable:

• Přijď dřív.
• Přijď dříve.

Both mean Come earlier.

In your sentence, dřív sounds very natural.

Yes, formally it looks like a present-tense form, but nastoupit is a perfective verb.

In Czech, perfective verbs do not normally have a true present meaning. Their present forms usually refer to the future.

So:

• nastoupit = to get on / board (perfective)
• nastoupíme = we will get on / once we get on

That is why Čím dřív nastoupíme... means The sooner we get on..., not The sooner we are getting on...

Compare:

• nastupujeme = we are getting on / we get on regularly
  imperfective: nastupovat
• nastoupíme = we will get on
  perfective: nastoupit

Because the two verbs behave differently.

1. nastoupit is perfective, so its present form can express the future:

   • nastoupíme = we will get on

2. být is not used that way. To say future with to be, Czech uses:

   • budu, budeš, bude, budeme...

So:

• nastoupíme = we will board
• budeme v centru = we will be in the center

That is why the sentence mixes:

• Čím dřív nastoupíme...
• tím dřív budeme...

Because nastoupit do is the standard pattern for boarding an enclosed vehicle in Czech.

So you say:

• nastoupit do tramvaje = get on the tram
• nastoupit do vlaku = get on the train
• nastoupit do autobusu = get on the bus
• nastoupit do auta = get into the car

The preposition do takes the genitive case, which is why tramvaj changes to tramvaje.

It is genitive singular.

The dictionary form is:

• tramvaj

After do, you need the genitive:

• do tramvaje

This is because do normally expresses movement into something and requires the genitive.

Because v centru means in the center / at the center, which describes the resulting location, not movement toward it.

Compare:

• Pojedeme do centra. = We’ll go to the center.
  movement toward a destination
• Budeme v centru. = We’ll be in the center.
  location

In your sentence, the second half says where you will be:

• budeme v centru = we’ll be in the center

So v is correct.

It is the locative singular of centrum.

After v when it means in, Czech uses the locative:

• v centru = in the center
• v domě = in the house
• v Praze = in Prague

So:

• centrum = dictionary form
• v centru = locative singular

Yes, Czech word order is flexible, but not random.

Here:

• Čím dřív nastoupíme do tramvaje, tím dřív budeme v centru.

the adverb dřív comes early because it is the key comparative idea: the sooner.

You could sometimes hear similar variations depending on emphasis, but this version is very natural and standard. Keeping dřív right after čím and tím makes the parallel structure very clear:

• Čím dřív ..., tím dřív ...

That symmetry is one reason this word order sounds so good.

You could use that as a grammar note for yourself, but it is not how you would normally translate it into natural English.

The natural English equivalent is:

• The sooner we get on the tram, the sooner we’ll be in the center.

A very literal translation can help show the structure, but for actual meaning, it is better to remember:

• čím ... tím ... = the ... the ...

It is singular.

• tramvaj = a tram
• do tramvaje = onto/into the tram

Even though tramvaje can also be a plural form in other contexts, here the preposition do and the meaning make it clear that this is genitive singular.

Not in natural English. Czech allows this compact structure, but English needs the correlative form:

• The sooner we get on the tram, the sooner we’ll be in the center.

So even though the Czech is short and efficient, you should not copy its structure too literally into English.

Yes, very common.

The čím ... tím ... pattern is used all the time in everyday Czech. It is worth memorizing as a whole structure.

Some useful examples:

• Čím víc, tím líp. = The more, the better.
• Čím dřív, tím líp. = The sooner, the better.
• Čím déle čekáš, tím je to horší. = The longer you wait, the worse it is.
• Čím víc pracuje, tím víc vydělá. = The more he works, the more he earns.

So this sentence is a very good model for a common Czech pattern.