Varasin ajan opettajalle kysyäkseni, miten voisin löytää paremman harjoittelupaikan ensi lukuvuodeksi.

Varasin is the 1st person singular past tense of varata.

• varata = to book, reserve
• varasin = I booked / I reserved

So the sentence starts with I booked...

Varata aika is a very common Finnish expression meaning to book an appointment.

It is not usually understood literally as reserve time in normal conversation. It is the standard thing you say for appointments with a teacher, doctor, hairdresser, official, and so on.

So:

• varasin ajan = I booked an appointment

Because aika is the object of the verb varata, and here it is a total object: the appointment was fully booked as a completed action.

In singular, the total object often looks like the genitive form:

• aika → ajan

So:

• varasin ajan = I booked the appointment / an appointment

This is very common in Finnish:

• luin kirjan = I read the book
• ostin auton = I bought a car
• varasin ajan = I booked an appointment

Grammatically, yes, -lle is the allative ending and often means to or onto. But with varata aika jollekulle, Finnish uses this case to mean book an appointment with someone / to see someone.

So:

• varata aika opettajalle = to book an appointment with the teacher

This is the same pattern as:

• varata aika lääkärille = book an appointment with the doctor
• varata aika hammaslääkärille = book an appointment with the dentist

So it does not mean you booked an appointment for the teacher to use. It means the teacher is the person you arranged to meet.

Kysyäkseni is a special infinitive form meaning to ask / in order to ask, with the subject built in.

It breaks down like this:

• kysyä = to ask
• -kse- = a purpose marker
• -ni = my / I

So kysyäkseni means roughly:

• in order for me to ask
• more naturally in English: to ask

In this sentence:

• Varasin ajan opettajalle kysyäkseni...
• I booked an appointment with the teacher to ask...

This form is common in somewhat careful or written Finnish.

Because Finnish often uses -kse- + possessive suffix to show purpose when the subject of both actions is the same.

Here, the same person:

• booked the appointment
• wanted to ask something

So kysyäkseni neatly means so that I could ask or to ask.

A more explicit alternative would be something like:

• Varasin ajan opettajalle, jotta voisin kysyä...

That also works, but kysyäkseni is shorter and a bit more formal or written in style.

Voisin is the conditional form of voida.

• voin = I can
• voisin = I could / I might be able to

So:

• miten voin löytää = how can I find
• miten voisin löytää = how could I find / how might I find

Here the conditional sounds softer and more natural because the speaker is asking for advice or possibilities, not stating a simple direct ability question.

It gives the sentence a meaning like:

• how I could find
• how I might find

Because this is the object of löytää.

The basic form is:

• parempi harjoittelupaikka = a better internship placement / training place

But after löytää, in this sentence, it becomes a total object in the singular, so both words change form:

• parempi → paremman
• harjoittelupaikka → harjoittelupaikan

So:

• löytää paremman harjoittelupaikan = to find a better internship placement

The idea is that the speaker wants to find one specific better place, as a complete result.

It is a compound word:

• harjoittelu = internship, practical training, work placement, teaching practice, etc.
• paikka = place

So harjoittelupaikka literally means training place or internship place, but in natural English it often means:

• internship placement
• placement
• training placement
• sometimes practice school/workplace, depending on context

The exact English translation depends on what kind of studies the person is doing.

Because -ksi here expresses for a future period or for use in that period.

• ensi lukuvuodeksi = for next school year / for the next academic year
• ensi lukuvuonna = during next school year

That is an important difference.

In this sentence, the speaker wants to find a better placement for the next school year, meaning a placement intended to be used then.

Compare:

• Etsin asuntoa ensi vuodeksi. = I am looking for an apartment for next year.
• Asun Suomessa ensi vuonna. = I will live in Finland next year.

So lukuvuodeksi is very natural here.

Yes. In standard Finnish, a comma is normally used before a subordinate clause, including an embedded question.

Here miten voisin löytää paremman harjoittelupaikan ensi lukuvuodeksi is an embedded question: it is the content of what the speaker wanted to ask.

So the comma is correct:

• ...kysyäkseni, miten voisin löytää...

This is similar to English punctuation in sentences like:

• I asked how I could find...

The word order is fairly natural and neutral, but Finnish word order is more flexible than English word order.

This sentence is well-formed as it stands:

• Varasin ajan opettajalle kysyäkseni, miten voisin löytää paremman harjoittelupaikan ensi lukuvuodeksi.

You could also hear small variations depending on emphasis, for example putting opettajalle earlier or later. But the given version sounds normal and clear.

The main thing is that the relationships are shown by the case endings:

• opettajalle tells you the teacher is the person the appointment is with
• harjoittelupaikan shows the object
• lukuvuodeksi shows for next school year

So Finnish can move parts around more easily than English, though not every order sounds equally natural.