Αυτή την εβδομάδα είμαι πολύ πιεσμένος στη δουλειά.

Because αυτή την εβδομάδα is a time expression meaning this week.

In Greek, expressions of time like this week, this month, this year often use the accusative case:

• αυτή η εβδομάδα = this week as the subject
• αυτή την εβδομάδα = this week meaning during this week

So in this sentence, it means during this week, I am very pressured at work.

Because εβδομάδα is a feminine noun in the accusative singular, and the demonstrative/adjective has to match it.

So:

• nominative: αυτή η εβδομάδα
• accusative: αυτή την εβδομάδα

The article changes from η to την, while αυτή stays the same in spelling.

Πιεσμένος means pressured, stressed, or under pressure.

It comes from the verb πιέζω = to press / pressure, but here it is being used as an adjective.

So:

• είμαι πιεσμένος = I am pressured / stressed
• not necessarily emotionally upset, but often busy, overloaded, under pressure

In this sentence, the idea is especially under pressure because of work.

Greek very often uses είμαι + adjective where English uses expressions like I’m stressed.

So:

• είμαι πιεσμένος = I am stressed / under pressure
• literally, it works like an adjective describing your state

This is very natural Greek.

Yes. It must agree with the speaker’s gender and number.

For one person:

• masculine: πιεσμένος
• feminine: πιεσμένη
• neuter: πιεσμένο (not used for a person in this context)

Examples:

• a man says: είμαι πολύ πιεσμένος
• a woman says: είμαι πολύ πιεσμένη

For plural:

• είμαστε πολύ πιεσμένοι / πιεσμένες

Because Greek usually does not need the subject pronoun when the verb already shows the person.

Είμαι already means I am, so εγώ is unnecessary unless you want emphasis.

Compare:

• είμαι πολύ πιεσμένος = I’m very pressured
• εγώ είμαι πολύ πιεσμένος = I’m the one who’s very pressured / extra emphasis

Πολύ modifies the adjective πιεσμένος.

So:

• πολύ πιεσμένος = very pressured / very stressed

This is the normal position:

• είμαι πολύ πιεσμένος

Στη is just the contracted form of σε τη, which in standard Greek becomes στη(ν).

So:

• στη δουλειά = at work
• στην δουλειά is also seen in writing, but στη δουλειά is the usual form before a consonant

Both refer to:

• στη δουλειά = at work / in my work

Here it means at work, in my job, or because of work.

Greek στη δουλειά can cover several close meanings depending on context:

• location: at work
• sphere/context: in my work
• source of pressure: with work / because of work

In this sentence, the natural English meaning is something like I’m very pressured at work this week or I’m very busy/stressed with work this week.

Not necessarily.

Greek often says simply στη δουλειά when it is obvious that it means at work / in one’s job.

So:

• στη δουλειά = perfectly natural
• στη δουλειά μου = more explicit, at my work / in my job

Both are correct, but the version without μου sounds very natural.

Yes, and that is also very natural, but it means something slightly different.

• Αυτή την εβδομάδα είμαι πολύ πιεσμένος στη δουλειά
  = This week I’m very pressured/stressed at work

• Αυτή την εβδομάδα έχω πολλή δουλειά
  = This week I have a lot of work

The second focuses more on quantity of work.
The original sentence focuses more on your state: you feel under pressure.

Yes. Greek word order is fairly flexible.

The original sentence:

• Αυτή την εβδομάδα είμαι πολύ πιεσμένος στη δουλειά

Other natural possibilities:

• Είμαι πολύ πιεσμένος στη δουλειά αυτή την εβδομάδα
• Στη δουλειά είμαι πολύ πιεσμένος αυτή την εβδομάδα

The original puts this week first, which gives that time frame a little prominence.

It is neutral and natural. You could say it in everyday speech, and it would also be fine in many normal written contexts.

It is not slang, and it is not especially formal either.

The sentence itself does not need to change much, because it is about your own condition, not about the other person.

You would still say:

• Αυτή την εβδομάδα είμαι πολύ πιεσμένος στη δουλειά.

If the speaker is female:

• Αυτή την εβδομάδα είμαι πολύ πιεσμένη στη δουλειά.

Politeness in Greek usually affects how you address the other person, not this kind of self-description.