Varasin ajan optikolle, koska haluan tarkistuttaa silmäni ensi viikolla.

Varasin is the past tense of varata, meaning to book / reserve.

• varata = to book
• varasin = I booked

So the sentence starts in the past: I booked an appointment...

If it were present tense, it would be varaan = I book / I am booking.

Because aika is the object here, and in this kind of sentence it appears as ajan.

There are two useful things to know:

• aika normally means time
• in the expression varata aika, it often means an appointment / a time slot

So varasin ajan means I booked an appointment.

Grammatically, ajan is the total object form here. In an affirmative sentence with a completed singular object, Finnish often uses this genitive-looking form:

• varasin ajan = I booked the appointment / an appointment

Optikolle is the allative form of optikko (optician), and the ending -lle often means to / for / onto.

Here it marks the person with whom the appointment is booked:

• varata aika optikolle = to book an appointment with the optician

So:

• optikko = optician
• optikolle = to/for the optician

This is just how Finnish normally expresses this idea.

Here koska means because.

So:

• koska haluan tarkistuttaa silmäni = because I want to have my eyes checked

Be careful, because koska can also mean when in questions:

• Koska tulet? = When are you coming?

But in this sentence, it is clearly the conjunction because.

After haluta (to want), Finnish normally uses the next verb in the first infinitive, which is the dictionary form.

So:

• haluan mennä = I want to go
• haluan syödä = I want to eat
• haluan tarkistuttaa = I want to have checked

That is why you get:

• haluan tarkistuttaa silmäni = I want to have my eyes checked

This is a very important difference.

• tarkistaa = to check
• tarkistuttaa = to have something checked (by someone else)

So:

• Tarkistan paperin. = I check the paper myself.
• Tarkistutan silmäni. = I have my eyes checked by a professional.

In this sentence, the speaker is not personally checking their own eyes. They are going to an optician, who will do it. That is why tarkistuttaa is the natural choice.

Silmäni means my eye or my eyes, depending on context.

The ending -ni is the possessive suffix meaning my:

• silmä = eye
• silmäni = my eye / my eyes

In Finnish, forms with a possessive suffix can sometimes be ambiguous between singular and plural. Here the meaning is understood as my eyes, because that is what people normally have checked by an optician.

So even though English needs eyes, Finnish can simply use silmäni and let the context make it clear.

Silmiäni would be the partitive plural form.

In this sentence, Finnish uses silmäni because the eyes are treated as a whole, complete object of the action: the speaker wants to have their eyes checked as a complete thing.

Very roughly:

• silmäni = my eye(s), as a whole
• silmiäni = my eye(s), in a more partial / ongoing / less bounded sense

With tarkistuttaa silmäni, the idea is a normal complete eye check, so silmäni is the natural form.

Because Finnish often uses a case ending for time expressions, and here viikolla is the adessive form.

• viikko = week
• viikolla = on/during the week

So:

• ensi viikolla = next week

Literally, it is somewhat like on next week, but in natural English we just say next week.

This pattern is very common in Finnish time expressions.

Ensi means next or upcoming, especially the very next one in time.

So:

• ensi viikolla = next week
• ensi vuonna = next year
• ensi kerralla = next time

It is a very common word in everyday Finnish.

Yes. In standard Finnish, a subordinate clause introduced by a word like koska is normally separated with a comma.

So this is correct:

• Varasin ajan optikolle, koska haluan tarkistuttaa silmäni ensi viikolla.

The comma works much like in English before because-clauses, although the exact rules are not always identical.

The given word order is natural and neutral, but Finnish word order is fairly flexible.

This sentence is the most straightforward version:

• Varasin ajan optikolle, koska haluan tarkistuttaa silmäni ensi viikolla.

You could move some parts for emphasis, for example:

• Ensi viikolla haluan tarkistuttaa silmäni, joten varasin ajan optikolle.

That changes the focus a bit, but the core meaning stays similar.

So the original order is not the only possible one, but it is a very normal way to say it.