Herään aikaisin ehtiäkseni kouluun koeviikon ensimmäisenä päivänä.

Herään is the 1st person singular present tense of herätä, to wake up. So grammatically it means I wake up.

In a sentence like this, the Finnish present tense can describe:

• a habit: I wake up early
• a planned future action: I’ll wake up early

Without more context, it most naturally sounds like a regular or intended action.

Aikaisin is an adverb, so it means early in the sense of waking up early.
Aikainen is an adjective, used with nouns, for example aikainen aamu = an early morning.

So:

• Herään aikaisin = I wake up early
• aikainen päivä = an early day or an early date/time, depending on context

Ehtiäkseni means in order for me to make it / to be in time.

It comes from the verb ehtiä, which often means:

• to have enough time
• to manage to get somewhere in time
• to make it

So in this sentence, ehtiäkseni kouluun is not just to get to school, but more specifically to make it to school in time.

It is a purpose form built from the verb plus -kse- and a possessive ending.

Here:

• ehtiä = the verb
• -kse- = marks purpose, like in order to
• -ni = my / I

So ehtiäkseni literally works like for me to have time / in order for me to make it.

This construction is very common in Finnish when the subject of both actions is the same:

• Herään aikaisin ehtiäkseni kouluun.
  I wake up early to make it to school.

Compare:

• Opiskelen paljon läpäistäkseni kokeen.
  I study a lot in order to pass the exam.

Because kouluun is the illative case of koulu, and the illative is used for movement to/into something.

So:

• koulu = school
• kouluun = to school / into school

Since the idea is reaching a destination, Finnish uses kouluun.

Not quite. The verb ehtiä adds the idea of having enough time or not being late.

So ehtiäkseni kouluun is closer to:

• to make it to school
• to get to school in time

If you only wanted to say to go to school, you would normally use mennä kouluun.

Koeviikon is the genitive singular of koeviikko, which means exam week.

The genitive here gives the meaning of exam week:

• koeviikon ensimmäinen päivä = the first day of exam week

So koeviikon tells you which first day we are talking about.

Also, koeviikko itself is a compound:

• koe = exam/test
• viikko = week

Because Finnish uses the essive case for many time expressions, and here it means on the first day.

So:

• ensimmäinen päivä = the first day
• ensimmäisenä päivänä = on the first day

Both words change because the adjective agrees with the noun:

• ensimmäinen → ensimmäisenä
• päivä → päivänä

This is similar to other time expressions such as:

• maanantaina = on Monday
• viimeisenä päivänä = on the last day

It means on the first day of exam week.

Its parts are:

• koeviikon = of exam week
• ensimmäisenä = first in the essive case
• päivänä = day, also in the essive case

The head word is päivänä. The rest describes which day it is.

So the structure is basically:

• [of exam week] [on the first] [day]

That sounds unusual in English if translated word for word, but it is normal Finnish structure.

No, Finnish word order is fairly flexible. The given order is natural, but you can move parts around for emphasis.

For example:

• Herään aikaisin ehtiäkseni kouluun koeviikon ensimmäisenä päivänä.
• Koeviikon ensimmäisenä päivänä herään aikaisin ehtiäkseni kouluun.

Both are fine. The second version puts more focus on when this happens.

Because ehtiä can already work with a destination. It often means to manage to get somewhere in time.

So:

• ehdin kouluun = I make it to school / I get to school in time

That is why Finnish does not need a separate mennä here. The idea of reaching school is already built into the combination of ehtiä + destination.

Yes. A very common alternative is a full purpose clause, for example:

• Herään aikaisin, jotta ehdin kouluun koeviikon ensimmäisenä päivänä.

This also means I wake up early so that I can make it to school on the first day of exam week.

The version with ehtiäkseni is more compact and elegant, while the jotta version may feel more straightforward to learners.