Két nappal ezelőtt érkeztem.

A natural breakdown is:

• két = two
• nappal ezelőtt = days ago / literally before this by two days
• érkeztem = I arrived

So the whole sentence means I arrived two days ago.

In Hungarian, két is normally used directly before a noun:

• két nap = two days
• két ember = two people

Kettő is more often used when the number stands alone:

• Hányan vannak? Kettő. = How many are there? Two.

So before nap, két is the normal form.

Because Hungarian often uses the suffix -val/-vel in expressions with ezelőtt to mean ago.

So:

• nap = day
• nap + -val = nappal

Then:

• két nappal ezelőtt = two days ago

This pattern is very common:

• három évvel ezelőtt = three years ago
• egy órával ezelőtt = an hour ago
• öt perccel ezelőtt = five minutes ago

This is because of assimilation. When -val/-vel is added to a word ending in a consonant, the v usually changes to match the final consonant of the word, and that consonant gets doubled in writing.

So:

• nap + val → nappal
• hét + vel → héttel
• év + vel → évvel
• perc + vel → perccel

So nappal is the regular result of adding -val to nap.

Ezelőtt means ago in expressions like this.

It is historically related to ez = this and előtt = before/in front of, so the literal sense is something like before this. But in modern Hungarian, in this pattern, it simply means ago:

• két nappal ezelőtt = two days ago
• egy héttel ezelőtt = a week ago

That is the normal Hungarian structure for this kind of expression:

[amount of time] + ezelőtt

Examples:

• két nappal ezelőtt = two days ago
• három hónappal ezelőtt = three months ago
• sok évvel ezelőtt = many years ago

So the order is different from English. English says two days ago, and Hungarian also puts the ago-word after the time expression.

Érkeztem is the past tense, first person singular form of érkezni = to arrive.

It can be understood as:

• érkez- = verb stem
• -t- = past marker
• -em = I ending

So érkeztem means I arrived.

Hungarian verbs usually show the subject clearly, so you do not need a separate pronoun.

Usually no. In Hungarian, the verb ending already tells you who the subject is.

• érkeztem = I arrived
• érkeztél = you arrived
• érkezett = he/she arrived

So én is optional. You add it mainly for emphasis or contrast:

• Én érkeztem, nem ő. = I arrived, not him/her.

Without emphasis, érkeztem by itself is perfectly normal.

Yes, Hungarian word order is flexible, but the sentence as given is the most neutral and natural if you are simply stating when you arrived:

• Két nappal ezelőtt érkeztem. = neutral: I arrived two days ago.

You may also hear:

• Érkeztem két nappal ezelőtt.

This is possible, but it sounds more marked and is less neutral in many contexts. For a learner, the original order is the safest choice.

Yes. Két napja érkeztem is also a common way to say I arrived two days ago.

The two versions are close in meaning:

• Két nappal ezelőtt érkeztem.
• Két napja érkeztem.

A rough difference is:

• két nappal ezelőtt points to a past time point
• két napja emphasizes that it has been two days since the arrival

In everyday speech, both are very natural.

Normally, no, not for two days ago.

Előtt by itself means before or in front of, and it is used differently:

• ebéd előtt = before lunch
• a ház előtt = in front of the house

For ago, Hungarian normally uses:

• két nappal ezelőtt or
• két napja

So két nap előtt is not the standard way to say two days ago.

Yes, nappal can also mean by day / in the daytime, because that is another valid form in Hungarian.

For example:

• Nappal dolgozom, éjjel alszom. = I work by day, I sleep at night.

But in két nappal ezelőtt, the meaning is clearly day + -val, so it means two days ago, not in the daytime. The surrounding words make the meaning clear.