Mesmo que eu esteja cansado, vou estudar na mesma.

Because mesmo que normally triggers the subjunctive.

In this sentence, esteja is the present subjunctive form of estar for eu. Portuguese uses the subjunctive after expressions like mesmo que when the speaker is talking about a condition, possibility, or concession rather than stating a simple fact.

So:

• Mesmo que eu esteja cansado... = Even if I am tired...
• Eu estou cansado = I am tired (plain statement of fact, indicative)

A learner often wants to say mesmo que eu estou cansado, but that is not correct here.

Mesmo que means even if.

It introduces a concessive idea: something will happen despite the condition mentioned.

So the structure is:

• Mesmo que + subjunctive

Examples:

• Mesmo que chova, vou sair. = Even if it rains, I’m going out.
• Mesmo que ele não venha, começamos. = Even if he doesn’t come, we’ll start.

In your sentence, the idea is:

• Even if I’m tired, I’m still going to study.

Esteja is the present subjunctive of estar.

Here is the full present subjunctive of estar:

• eu esteja
• tu estejas
• ele/ela/você esteja
• nós estejamos
• vós estejais
• eles/elas/vocês estejam

This tense is often used after:

• expressions of doubt
• wishes
• emotions
• impersonal expressions
• conjunctions like mesmo que, embora, para que, antes que

Because cansado agrees with the speaker.

In Portuguese, adjectives usually agree in gender and number with the person or thing they describe.

So:

• a male speaker says eu esteja cansado
• a female speaker says eu esteja cansada

Plural examples:

• Mesmo que estejamos cansados = if the group is all male or mixed
• Mesmo que estejamos cansadas = if the group is all female

Vou estudar is the near future / periphrastic future: literally I am going to study.

It is formed with:

• ir in the present tense + infinitive

So:

• vou estudar = I’m going to study / I will study

In everyday Portuguese, this structure is extremely common and often more natural in speech than the simple future:

• vou estudar instead of estudarei

Both can mean I will study, but vou estudar is very common and natural in both speech and writing.

Here, na mesma means something like:

• all the same
• anyway
• still
• even so

So:

• vou estudar na mesma = I’m going to study anyway / all the same

In European Portuguese, na mesma is a very common informal expression.

It often adds the idea:

• despite that
• it doesn’t change the outcome

Examples:

• Está a chover, mas vou na mesma. = It’s raining, but I’m going anyway.
• Ele não gosta, mas eu faço na mesma. = He doesn’t like it, but I’ll do it anyway.

Yes, it is especially common in European Portuguese.

A learner of Portugal Portuguese should definitely recognize and use it. In Brazilian Portuguese, people may understand it, but other expressions such as mesmo assim or de qualquer maneira may sound more neutral depending on the context.

For Portugal Portuguese, na mesma is very natural in everyday speech.

Sometimes yes, but they are not always identical.

In this sentence:

• vou estudar na mesma sounds very natural in European Portuguese.
• vou estudar mesmo assim is also possible and means something like I’ll study even so / even then.

The difference is often one of style and emphasis:

• na mesma = very colloquial, very common in Portugal, anyway / all the same
• mesmo assim = even so / even then, sometimes slightly more explicit

So both can work, but na mesma is a great phrase to learn for Portugal Portuguese.

Yes. Portuguese often drops subject pronouns when the verb form already makes the subject clear.

So both are possible:

• Mesmo que eu esteja cansado, vou estudar na mesma.
• Mesmo que esteja cansado, vou estudar na mesma.

Because esteja could in some contexts also refer to ele/ela/você, keeping eu can make the sentence clearer. So including eu is perfectly natural here.

The sentence has two parts:

1. Mesmo que eu esteja cansado
   This is the subordinate concessive clause: Even if I’m tired

2. vou estudar na mesma
   This is the main clause: I’m going to study anyway

This order is very natural because the speaker first introduces the obstacle or condition, then gives the main result.

You could also focus more on the main clause in other contexts, but this version is the most straightforward one.

Not with exactly the same meaning.

• Mesmo que = even if
• embora = although / even though

That is an important difference.

Compare:

• Mesmo que eu esteja cansado, vou estudar.
  = Even if I’m tired, I’ll study.
  This can refer to a possible condition.

• Embora eu esteja cansado, vou estudar.
  = Although I’m tired, I’ll study.
  This sounds more like the tiredness is an actual fact.

Both take the subjunctive, but they express slightly different ideas.

It is mostly neutral, but na mesma gives it an informal, conversational feel.

The rest of the sentence is completely standard:

• Mesmo que eu esteja cansado, vou estudar...

If you wanted a more formal ending, you could say:

• Mesmo que eu esteja cansado, vou estudar de qualquer forma.
• Mesmo que eu esteja cansado, vou estudar mesmo assim.

But for natural European Portuguese conversation, na mesma is excellent.

You would change both the verb and the adjective as needed.

Female speaker:

• Mesmo que eu esteja cansada, vou estudar na mesma.

With tu:

• Mesmo que estejas cansado/cansada, vais estudar na mesma.

With ele:

• Mesmo que ele esteja cansado, vai estudar na mesma.

With nós:

• Mesmo que estejamos cansados/cansadas, vamos estudar na mesma.

This is a good reminder that in Portuguese you often have to watch:

1. the subjunctive verb form
2. the adjective agreement

A close word-by-word breakdown is:

• Mesmo que = even if
• eu = I
• esteja = am / be in the present subjunctive
• cansado = tired
• vou estudar = I am going to study
• na mesma = all the same / anyway

So the full sense is:

Even if I’m tired, I’m going to study anyway.