Ég fer til augnlæknis á morgun, því síðasta sjónpróf var fyrir löngu síðan.

In Icelandic, the present tense is very often used for the near future when the time is clear from context.

So:

• Ég fer ... á morgun = I’m going ... tomorrow / I’ll go ... tomorrow

This is very natural Icelandic.
You can say Ég mun fara ... for I will go ..., but that usually sounds more explicit, more formal, or less conversational in a sentence like this.

Because til governs the genitive case in Icelandic.

The dictionary form is:

• augnlæknir = eye doctor

After til, it becomes genitive singular:

• til augnlæknis = to an/the eye doctor

So this is a very important pattern to learn:

• til læknis = to a doctor
• til tannlæknis = to a dentist
• til augnlæknis = to an eye doctor

Icelandic does not have an indefinite article like English a/an.

So where English says:

• to an eye doctor

Icelandic simply says:

• til augnlæknis

Whether English uses a, an, or sometimes the, Icelandic often just uses the bare noun and lets context do the work.

Yes. In English, we often say go to the doctor / go to the dentist / go to the eye doctor with the, even when we do not mean one very specific doctor.

Icelandic often does not use the definite form in this kind of expression. So:

• til augnlæknis

can naturally correspond to English to the eye doctor.

If you really wanted to emphasize a specific eye doctor, Icelandic could use the definite form, but in an ordinary sentence like this, til augnlæknis is the normal choice.

Á morgun means tomorrow.

It is a fixed expression, and you should learn it as a chunk.

A useful contrast is:

• á morgun = tomorrow
• í morgun = this morning

So the preposition changes the meaning:

• á
  • morgun → tomorrow
• í
  • morgun → this morning

Here, því means because / since.

It introduces the reason:

• Ég fer til augnlæknis á morgun = I’m going to the eye doctor tomorrow
• því síðasta sjónpróf var fyrir löngu síðan = because my last eye test was a long time ago

This því is a conjunction. It is fairly common in written and careful spoken Icelandic.

A very common alternative is:

• af því að = because

So the sentence could also be phrased with af því að, though the version with því is perfectly correct.

Because Icelandic punctuation uses commas before many subordinate or explanatory clauses more often than English does.

So in Icelandic, a comma before því in a sentence like this is standard:

• Ég fer ..., því ...

If you are an English speaker, this may feel slightly more comma-heavy than English, but it is normal Icelandic style.

Because síðasta is an adjective meaning last, and it has to agree with the noun sjónpróf.

Here:

• sjónpróf is neuter singular
• so the adjective appears as síðasta

By contrast, síðast is usually an adverb, meaning lastly or most recently, not an adjective modifying a noun.

So:

• síðasta sjónpróf = the last eye test / my last eye test

is correct.

Words like síðasti / síðasta / síðasta often behave as if they already contain a definite idea, like the last or my last in English.

So:

• síðasta sjónpróf

very naturally means:

• the last eye test
• or, in context, my last eye test

This is normal Icelandic usage. You do not need a separate word for the here.

It is nominative singular neuter.

Why?

Because sjónpróf is the subject of the verb var:

• síðasta sjónpróf var ...
• the last eye test was ...

Since it is the subject, nominative is exactly what you would expect.

The whole phrase means a long time ago.

You can think of it roughly as an idiomatic expression rather than translating each word separately.

Parts of it:

• fyrir löngu = long ago
• síðan can also help express ago in this kind of context

Together:

• fyrir löngu síðan = a long time ago

This combination is very common and natural. English learners sometimes feel it is redundant because both parts seem time-related, but in Icelandic this full phrase is idiomatic.

Löngu is the dative singular form of the adjective langur (long).

It appears here because of the preposition fyrir, which in this expression takes the dative:

• fyrir löngu

You do not need to build this phrase from scratch every time; it is best learned as a fixed expression meaning long ago.

Yes, you could, but it changes the feel a little.

• Ég fer til augnlæknis á morgun = simple, neutral, very natural
• Ég er að fara til augnlæknis á morgun = more like I’m going / I’m about to go / I’ve got plans to go

The original sentence is the most straightforward and natural version for a general statement about tomorrow’s plan.