Eve vardım mı sana haber veririm.

This is one of the most confusing but very common patterns in Turkish.

In Eve vardım mı, sana haber veririm, the form -dım mı does not mean a normal past-tense question here. Together, it creates a time clause meaning:

• when I arrive
• once I have arrived
• as soon as I arrive

So vardım mı is understood as something like once I get home.

This pattern is very common in Turkish:

• Eve geldim mi uyurum. → Once I get home, I sleep / I go to sleep.
• Seni gördüm mü mutlu oluyorum. → Whenever I see you, I become happy.

So even though vardım looks like I arrived, the whole structure vardım mı functions differently.

Formally, yes, it is the same particle as the yes/no question particle mı / mi / mu / mü. But in this sentence, it is not asking a question.

Here it combines with a verb form to make a temporal meaning:

• geldim mi → when I come / once I come
• bitti mi → when it is finished / once it is finished
• vardım mı → when I arrive / once I arrive

So in this sentence, mı is part of a fixed grammatical pattern, not a separate question like Did I arrive?

Because the verb varmak normally goes with the dative case when it means to arrive at / reach a place.

• ev = house / home
• eve = to home / homeward / to the house
• evde = at home / in the house

So:

• eve varmak = to arrive home
• evde olmak = to be at home

That is why eve vardım mı means when I arrive home.

Vardım comes from varmak, which means to arrive, to reach, or to get to a destination.

A useful comparison:

• gitmek = to go
• gelmek = to come
• varmak = to arrive / reach

So:

• Eve gidiyorum. → I am going home.
• Eve geldim. → I came home.
• Eve vardım. → I arrived home.

In your sentence, the focus is not on the journey itself, but on the moment of arrival.

Haber vermek is a common Turkish expression meaning:

• to let someone know
• to inform someone
• to give news
• sometimes to notify

So:

• sana haber veririm = I’ll let you know
• literally: I give news to you

Breakdown:

• haber = news / information
• veririm = I give / I will give in this kind of context

As a whole, haber vermek works like one verb phrase.

Turkish often uses the aorist form for things like:

• habitual actions
• general truths
• promises
• future actions in subordinate-clause contexts

So veririm here sounds natural and means something like:

• I’ll let you know
• I let you know

In English we strongly prefer a future form, but Turkish often does not need the explicit future ending here.

Compare:

• Gelince ararım. → I’ll call when I come.
• Bitince söylerim. → I’ll tell you when it’s finished.

You could also hear haber vereceğim, but haber veririm is very natural in this pattern.

Because haber vermek takes the person receiving the information in the dative case.

• sen = you
• sana = to you

So:

• sana haber veririm = I’ll let you know
• literally: I give news to you

Other examples:

• Bana haber ver. → Let me know.
• Ona haber verdim. → I let him/her know.

Not very naturally.

A rough word-by-word breakdown is:

• eve = to home
• vardım mı = once I arrived / when I arrive
• sana = to you
• haber veririm = I give news / I let know

But natural English would be:

• When I get home, I’ll let you know.
• Once I arrive home, I’ll let you know.

So the Turkish grammar does not match English word for word, especially in vardım mı.

Yes, the subject is understood as I.

Turkish usually does not need subject pronouns because the verb ending already shows the person:

• vardım = I arrived
• veririm = I give / I will give

So ben is unnecessary unless you want emphasis or contrast.

For example:

• Ben eve vardım mı sana haber veririm.
  This adds emphasis, something like I’ll let you know when I get home.

But the normal, natural version usually leaves ben out.

Yes, absolutely. That is a very common alternative.

These are all natural:

• Eve vardım mı sana haber veririm.
• Eve varınca sana haber veririm.
• Eve vardığımda sana haber veririm.

They are close in meaning, but there are small differences in style and feel:

• varınca = very common, straightforward when I arrive
• vardığımda = a bit more neutral/formal, when I have arrived / when I arrive
• vardım mı = often gives a sense like once I get there or as soon as I get there

So your sentence is natural, but it is not the only way to express the idea.

The natural pause is after vardım mı:

• Eve vardım mı, sana haber veririm.

That is because eve vardım mı is the time clause, and sana haber veririm is the main clause.

So the structure is:

• When I get home, → eve vardım mı
• I’ll let you know. → sana haber veririm

In writing, a comma is often helpful, although Turkish punctuation can be more flexible than English.

Because the Turkish yes/no question particle is written as a separate word.

So:

• vardım mı
• not vardımmı

This is true even when it is part of a structure like this one.

The form changes by vowel harmony:

• mı
• mi
• mu
• mü

Here it is mı because it follows vardım, whose last vowel is a back unrounded vowel, matching the harmony pattern.