Eve vardım mı sıcak bir duş alırım.

Here it means when I get home, once I get home, or as soon as I get home.

A very literal breakdown is:

• eve = to home
• vardım = I arrived
• mı = question particle

But in this pattern, the whole chunk vardım mı does not function like a normal question. It works as a time expression meaning once/when/as soon as.

So Eve vardım mı sıcak bir duş alırım means something like:

• When I get home, I take a hot shower
• As soon as I get home, I take a hot shower

This is one of those Turkish patterns that looks strange if you translate it word for word.

Normally mı / mi / mu / mü is the yes-no question particle. But after a past-tense verb, it can also create a meaning like:

• when
• once
• as soon as
• sometimes whenever

So vardım mı here is a fixed-type structure, not a genuine question.

Also, mı is written separately because the Turkish question particle is always written separately:

• geldi mi
• yaptın mı
• vardım mı

And here it is mı because of vowel harmony: the last vowel in vardım is ı.

Because Turkish uses this special structure:

• verb + past tense + personal ending + mı/mi/mu/mü

In this structure, the past form does not simply mean past time in the usual way. Instead, it helps form a time clause meaning when/once/as soon as.

So:

• vardım mı does not mean only did I arrive?
• in this sentence it means once I arrive / when I get there

This is a good example of why Turkish tense forms do not always match English tense forms one-to-one.

Alırım is the aorist form of almak.

Here it often expresses:

• a habitual action
• something one typically does
• a regular consequence

So sıcak bir duş alırım is naturally understood as:

• I take a hot shower
• or, depending on context, I will take a hot shower

In this sentence, the aorist makes it sound like a usual routine:

• When I get home, I take a hot shower

If you wanted a clearer one-time future meaning, you might say:

• Eve varınca sıcak bir duş alacağım

That sounds more like When I get home, I will take a hot shower on one particular occasion.

Because varmak takes the dative case for the destination.

• ev = house / home
• eve = to home
• evde = at home
• evi = the house as an object

With movement toward a place, Turkish usually uses the dative:

• eve gitmek = to go home
• eve gelmek = to come home
• eve varmak = to arrive home

So eve vardım means I arrived home or more literally I arrived to home.

Yes, but the meaning changes slightly.

• varmak focuses on reaching the destination
• gitmek means to go
• gelmek means to come

So:

• Eve vardım mı... = Once I arrive home...
• Eve gittim mi... = less natural here, because gitmek focuses more on the act of going than on the moment of arrival
• Eve geldim mi... = also possible, meaning when I come home

In this sentence, the important idea is the moment of arrival, so varmak fits very well.

In Turkish, the usual order is:

• adjective + bir + noun

So:

• sıcak bir duş = a hot shower
• büyük bir ev = a big house
• güzel bir gün = a beautiful day

For English speakers, bir sıcak duş may feel tempting because English puts a before the adjective, but Turkish normally does not.

So sıcak bir duş is the natural order.

Duş almak is the standard Turkish expression for to take a shower.

Literally, it is:

• duş = shower
• almak = to take

So it works very much like English take a shower.

This is just a normal collocation in Turkish. Many everyday actions are expressed this way, with a noun plus a verb.

Also, duş appears without an accusative ending here because it is an indefinite object:

• bir duş almak
• sıcak bir duş almak

That is completely normal.

Most naturally, it sounds like a habit or usual routine:

• Whenever I get home, I take a hot shower
• Once I get home, I usually take a hot shower

That feeling comes mainly from alırım, the aorist.

However, in the right context, it can also refer to a single future situation, especially in conversational Turkish. Still, if you want to make the one-time future meaning clearer, Turkish often prefers something like:

• Eve varınca sıcak bir duş alacağım

So the original sentence leans more toward routine than single future plan.

Yes, absolutely. These are all possible, but they have slightly different feels.

• Eve vardım mı sıcak bir duş alırım.
  More conversational and punchy. Often feels like the moment I get home...

• Eve varınca sıcak bir duş alırım.
  Very common and neutral. A great everyday way to say it.

• Eve vardığımda sıcak bir duş alırım.
  More explicit and slightly more formal or careful.

All three can mean roughly the same thing in many contexts, but vardım mı has a more idiomatic spoken flavor.

The most natural order is exactly what you see:

• Eve vardım mı sıcak bir duş alırım.

Turkish word order is flexible, but the clause at the beginning sets the time frame first:

• When I get home, I take a hot shower

That makes the sentence easy and natural.

You could move things around in Turkish for emphasis, but that often sounds marked or less neutral. For a learner, the safest pattern is:

• time expression first
• main action second

So this sentence is a very good model to follow.