Τον επόμενο μήνα θα κάνω αίτηση για υποτροφία στο πανεπιστήμιο.

That phrase means next month, and it is being used as a time expression.

Greek often puts time expressions at the beginning of the sentence to set the scene first. So:

• Τον επόμενο μήνα = next month
• θα κάνω αίτηση... = I will apply...

You could also say:

• Θα κάνω αίτηση για υποτροφία τον επόμενο μήνα.

That is also correct. The original version just emphasizes when this will happen.

They are all in the masculine singular accusative and they agree with each other.

The basic dictionary form is:

• ο επόμενος μήνας = the next month

But in the sentence, Greek uses the accusative form for this time expression:

• τον επόμενο μήνα

So the changes are:

• ο → τον
• επόμενος → επόμενο
• μήνας → μήνα

This is normal adjective agreement in Greek: article + adjective + noun all match in gender, number, and case.

Because μήνα is the accusative singular form of μήνας.

• μήνας = nominative singular
• μήνα = accusative singular

Masculine nouns in -ας often lose the final -ς in the accusative singular.

So:

• ο μήνας = the month
• τον μήνα = the month as an object or in a time expression

θα marks the future.

So:

• κάνω = I do / I make
• θα κάνω = I will do / I will make

Greek forms the future very commonly with:

• θα + verb

In this sentence:

• θα κάνω αίτηση = I will apply / I will make an application

Because Greek usually leaves out subject pronouns when they are not needed.

The verb ending already tells you the subject:

• κάνω = I do / I make

So εγώ is unnecessary unless you want emphasis or contrast.

For example:

• Εγώ θα κάνω αίτηση, όχι αυτός. = I will apply, not him.

In your sentence, the subject is clear from κάνω, so no pronoun is needed.

Because κάνω αίτηση is a very common Greek expression meaning to apply or more literally to make/submit an application.

So:

• κάνω αίτηση = I apply / I submit an application

This is natural everyday Greek.

There is also a more formal verb-based option:

• υποβάλλω αίτηση = I submit an application

But κάνω αίτηση is very common and useful.

Yes. Αίτηση is the direct object of κάνω, so it is in the accusative.

The reason it looks unchanged is that many feminine nouns have the same form in the nominative and accusative singular.

So:

• nominative: η αίτηση
• accusative: την αίτηση

Without the article, the bare noun is still αίτηση in both cases.

That is why the form does not visibly change here.

Because Greek often leaves out the article when talking about something indefinite or general.

So:

• για υποτροφία = for a scholarship / for scholarship funding in a general sense

If you said:

• για μια υποτροφία = for a scholarship with a more explicitly indefinite sense
• για την υποτροφία = for the scholarship, meaning a specific one already known

In your sentence, για υποτροφία sounds natural and general.

Για takes the accusative.

So in:

• για υποτροφία

the noun is in the accusative. But again, with υποτροφία, the nominative and accusative singular look the same, so you do not see a form change.

This is very common in Greek:

• για δουλειά = for a job / for work
• για άδεια = for leave / for permission
• για υποτροφία = for a scholarship

Στο is a contraction of:

• σε + το = στο

The preposition σε can mean in, at, or to, depending on context.

So:

• στο πανεπιστήμιο can mean at the university, in the university, or sometimes to the university

In this sentence, the most natural English translation is usually at the university or to the university, depending on the intended meaning.

Because Greek uses the definite article more often than English does.

• στο πανεπιστήμιο literally = at/in/to the university

This usually suggests a specific university, or one that is understood from context.

If you said:

• σε πανεπιστήμιο

that would sound more like at a university or in university in a more general sense.

So στο πανεπιστήμιο is the natural choice if a particular university is meant.

It can feel slightly broad without extra context.

Most naturally, the sentence suggests something like:

• I will apply for a scholarship at the university
• or I will apply to the university for a scholarship

If you wanted to make university scholarship very explicit, Greek could say something like:

• για υποτροφία του πανεπιστημίου

That more clearly means for a scholarship of the university.

So the original sentence is natural, but context helps decide the exact nuance.

Yes. Greek word order is fairly flexible.

For example, these are all possible:

• Τον επόμενο μήνα θα κάνω αίτηση για υποτροφία στο πανεπιστήμιο.
• Θα κάνω αίτηση για υποτροφία στο πανεπιστήμιο τον επόμενο μήνα.
• Για υποτροφία θα κάνω αίτηση στο πανεπιστήμιο τον επόμενο μήνα.

They do not all emphasize the same thing, but they are grammatically possible.

The original sentence puts Τον επόμενο μήνα first to highlight the time.