Τον υπόλοιπο χρόνο διαβάζω ελληνικά στο σπίτι.

Because this phrase is in the accusative, not the nominative.

Here τον υπόλοιπο χρόνο works as a time expression, meaning something like for the rest of the time / the rest of the time. Greek often uses the accusative for expressions of duration or time extent.

• ο υπόλοιπος χρόνος = nominative, the remaining time as a subject
• τον υπόλοιπο χρόνο = accusative, used here adverbially

Both the article and the adjective agree with χρόνο:

• τον = masculine singular accusative
• υπόλοιπο = masculine singular accusative
• χρόνο = masculine singular accusative

Υπόλοιπος means remaining, rest, or left over.

So:

• ο υπόλοιπος χρόνος = the remaining time
• τον υπόλοιπο χρόνο = the rest of the time

It is an adjective, so it changes form to match the noun it describes.

In Greek, χρόνος is often used in the singular to mean time in a general sense.

So τον υπόλοιπο χρόνο is a normal, idiomatic way to say the rest of the time.

A learner might associate χρόνος with year, because in some contexts it can mean that too, but here it clearly means time, not year.

Διαβάζω is:

• 1st person singular
• present tense
• active voice

So it means I read, I am reading, or sometimes I study, depending on context.

Greek present tense does not always match English present forms one-to-one. The exact English translation depends on the situation.

It can potentially mean either, depending on context.

In Modern Greek, διαβάζω very often means:

• read
• but also study

For example:

• Διαβάζω ένα βιβλίο = I’m reading a book
• Διαβάζω για το μάθημα = I’m studying for class

With ελληνικά, many learners notice that it may sound like I study Greek as well as I read Greek. The wider context usually makes it clear.

Ελληνικά here means Greek as a language.

Names of languages in Greek are often expressed with the neuter plural form:

• ελληνικά = Greek
• αγγλικά = English
• γαλλικά = French

After verbs like learn, speak, study, or read, Greek often uses the language name without an article:

• Μιλάω ελληνικά = I speak Greek
• Μαθαίνω ελληνικά = I’m learning Greek
• Διαβάζω ελληνικά = I read/study Greek

So ελληνικά is completely normal here.

Because in Greek, names of languages are normally not capitalized.

So:

• ελληνικά = Greek
• αγγλικά = English

This is different from English, where language names are capitalized.

Στο is a contraction of:

• σε
  • το → στο

So:

• στο σπίτι literally = in/at the house
• idiomatically = at home

This is very common in Greek. Similar contractions include:

• σε + την → στη(ν)
• σε + τον → στον

Because Greek usually does not need an explicit subject pronoun.

The verb ending already shows the subject:

• διαβάζω = I read / I am reading

So εγώ is usually omitted unless you want emphasis or contrast:

• Εγώ διαβάζω ελληνικά, όχι ισπανικά. = I study Greek, not Spanish.

This is very normal in Greek.

Greek word order is fairly flexible, and the beginning of the sentence is often used for topic or emphasis.

Starting with Τον υπόλοιπο χρόνο highlights the time frame:

• Τον υπόλοιπο χρόνο διαβάζω ελληνικά στο σπίτι. = As for the rest of the time, I read/study Greek at home.

Other word orders are also possible, but they may sound slightly different in focus or emphasis.

Yes, to a large extent.

For example, these are possible:

• Διαβάζω ελληνικά στο σπίτι τον υπόλοιπο χρόνο.
• Στο σπίτι διαβάζω ελληνικά τον υπόλοιπο χρόνο.

The core meaning stays similar, but the emphasis changes:

• Τον υπόλοιπο χρόνο... emphasizes when
• Στο σπίτι... emphasizes where
• Διαβάζω ελληνικά... emphasizes the action first

So the given order is natural, but not the only possible one.