Προτιμώ ριγέ πουκάμισο για τη δουλειά.

Because Greek often drops the subject pronoun when it is already clear from the verb ending.

Here, προτιμώ already means I prefer, so εγώ is not necessary.

• Προτιμώ = I prefer
• Εγώ προτιμώ = I prefer too, but with extra emphasis on I

So if you add εγώ, it sounds more like I prefer... as opposed to someone else.

Προτιμώ is the 1st person singular present tense form of the verb προτιμώ, meaning to prefer.

So:

• προτιμώ = I prefer
• προτιμάς = you prefer
• προτιμά = he/she/it prefers
• προτιμούμε / προτιμάμε = we prefer

In this sentence, it describes a general present preference, not necessarily only something happening right now.

In Greek, the article is often left out when you are speaking in a general or non-specific way.

So Προτιμώ ριγέ πουκάμισο can mean something like:

• I prefer a striped shirt
• I prefer striped shirts / striped shirt as a choice

If you say Προτιμώ ένα ριγέ πουκάμισο, that sounds a bit more like you mean one striped shirt or a striped shirt as a more specific item.

Compare:

• Προτιμώ ριγέ πουκάμισο = general preference
• Προτιμώ ένα ριγέ πουκάμισο = I prefer a striped shirt / one striped shirt
• Προτιμώ το ριγέ πουκάμισο = I prefer the striped shirt

Ριγέ means striped, and it is an indeclinable adjective in Modern Greek. That means it usually does not change form for gender, number, or case.

So you can say:

• ριγέ πουκάμισο = striped shirt
• ριγέ μπλούζα = striped blouse
• ριγέ παντελόνια = striped trousers

The form ριγέ stays the same.

That is different from many regular Greek adjectives, which do change depending on the noun.

Πουκάμισο is a neuter noun, and in this sentence it is the direct object of προτιμώ.

The important point is that many neuter nouns have the same form in nominative and accusative singular.

So:

• nominative: το πουκάμισο
• accusative: το πουκάμισο

Because of that, even though it is the object here, the form still looks the same.

Both are possible, but τη δουλειά is very common in everyday Modern Greek.

The full accusative feminine article is την, but the final -ν is often dropped before many consonants in ordinary usage:

• τη δουλειά
• την δουλειά

Both can be correct, depending on style, pronunciation, and how formal or careful the speaker is being. In this sentence, τη δουλειά sounds very natural.

Also, δουλειά is feminine:

• nominative: η δουλειά
• accusative: τη(ν) δουλειά

Because the preposition για normally takes the accusative case.

So:

• για τη δουλειά = for work

That is why you get the accusative article τη(ν).

This is very common in Greek:

• για το σπίτι = for the house / for home
• για τον φίλο μου = for my friend
• για τη δουλειά = for work

Για τη δουλειά here means something like:

• for work
• for wearing to work
• as suitable for work

It expresses purpose or use.

If you said στη δουλειά, that would mean at work or to work/in work, which is a different idea.

Compare:

• Προτιμώ ριγέ πουκάμισο για τη δουλειά.
  = I prefer a striped shirt for work.

• Φοράω ριγέ πουκάμισο στη δουλειά.
  = I wear a striped shirt at work.

So για is the better choice when you mean suitability or intended use.

No, Greek word order is fairly flexible. The sentence you have is neutral and natural, but other orders are also possible depending on emphasis.

For example:

• Προτιμώ ριγέ πουκάμισο για τη δουλειά.
  Neutral: I prefer a striped shirt for work.

• Για τη δουλειά προτιμώ ριγέ πουκάμισο.
  Emphasis on for work.

• Ριγέ πουκάμισο προτιμώ για τη δουλειά.
  Emphasis on striped shirt.

So the order can change, but the original version is a very normal one.

Yes, Greek sometimes allows that, especially in descriptive or stylistic contexts.

So both of these are possible:

• ριγέ πουκάμισο
• πουκάμισο ριγέ

But ριγέ πουκάμισο is the more straightforward and common order for a learner to use.

Putting it after the noun can sound more marked, descriptive, or stylistically specific depending on the context.

A simple English-friendly approximation is:

• Προτιμώ = pro-ti-MO
• ριγέ = ree-YE
• πουκάμισο = poo-KA-mi-so
• για = ya
• τη δουλειά = tee thoo-lya

A few important pronunciation notes:

• γ before ε in ριγέ sounds soft, close to y in this word.
• δ in δουλειά is like th in this, not like English d.
• The stress matters:
  • προτιμώ
  • ριγέ
  • πουκάμισο
  • δουλειά

If you place the stress incorrectly, Greek can sound unnatural or even become a different word.