Ég er ekki með mikla matarlyst í dag.

A natural breakdown is:

• Ég = I
• er = am
• ekki = not
• með = part of the expression vera með, which often means to have
• mikla = much / great in a form that matches matarlyst
• matarlyst = appetite
• í dag = today

So the sentence is literally something like I am not with much appetite today, but the natural English meaning is I don’t have much appetite today.

Er comes from the verb vera, which means to be.

Some present-tense forms are:

• ég er = I am
• þú ert = you are
• hann / hún / það er = he / she / it is

So in this sentence, er is simply the I am form of vera.

Because Icelandic very often uses vera með in everyday speech to express having something, especially a condition, feeling, symptom, or temporary state.

So Ég er ekki með mikla matarlyst í dag is a very natural conversational way to say I don’t have much appetite today.

Icelandic also has the verb hafa = to have, and you may also hear:

• Ég hef ekki mikla matarlyst í dag

That is understandable and possible, but vera með is extremely common in spoken Icelandic.

Not in the usual simple sense of with.

In this sentence, með is part of the fixed expression vera með, which often means to have or to be experiencing something.

For example:

• Ég er með höfuðverk = I have a headache
• Hún er með hita = She has a fever

So if you translate með here as plain with, it can sound strange in English. It is better to understand the whole expression vera með as one unit.

In a normal Icelandic statement, ekki usually comes after the finite verb.

So:

• Ég er ekki ... = I am not ...

This is very standard Icelandic word order.

Compare:

• Ég er þreyttur = I am tired
• Ég er ekki þreyttur = I am not tired

So the placement of ekki here is completely regular.

Because mikla is the form of the adjective mikill that agrees with matarlyst in gender, number, and case.

Here:

• matarlyst is feminine
• it is singular
• in this expression, it is in the accusative

So the adjective must match, giving:

• mikla matarlyst

A useful comparison:

• mikil matarlyst = nominative
• mikla matarlyst = accusative

English does not change adjectives this way, but Icelandic does.

That is a very good question.

Yes, the ordinary preposition með normally takes the dative. But in vera með meaning to have, learners should think of it as a special expression rather than a normal with phrase.

In this construction, the thing being had is often treated like an accusative object, which is why you get:

• mikla matarlyst

The noun matarlyst itself looks the same in nominative and accusative, but the adjective mikla shows you that the phrase is accusative.

So this is one of those cases where the full expression matters more than the usual rule for the standalone preposition.

Matarlyst is a compound word:

• matur = food
• lyst = desire, liking, appetite

So the literal idea is something like desire for food.

In normal English, though, the best translation is simply appetite.

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

So Icelandic often just uses the bare noun:

• matarlyst = appetite or an appetite, depending on context

In this sentence, English also naturally uses the bare noun:

• I don’t have much appetite today

So this part matches English fairly well in meaning, even though the grammar is different.

Yes, you could, but the meaning is slightly different.

• Ég er ekki svangur / svöng í dag = I’m not hungry today
• Ég er ekki með mikla matarlyst í dag = I don’t have much appetite today

These are close, but not identical.

Svangur / svöng focuses on the physical feeling of hunger.

Matarlyst focuses more on appetite or desire to eat. A person may not feel like eating even if they have not eaten much.

Also remember that svangur changes for gender:

• svangur if the speaker is male
• svöng if the speaker is female

Í dag means today.

Yes, it is a very common and very useful expression. You can treat it as a set phrase when you are starting out.

Examples:

• Ég vinn í dag = I’m working today
• Hún kemur í dag = She is coming today
• Við erum þreytt í dag = We are tired today

So in your sentence, í dag simply adds the time: today.