Learn Hardcore Portuguese (Portugal): Hoje eu estou totalmente cansado depois do exame.

Hoje estou totalmente cansado depois do exame. – perfectly correct and very natural.
Hoje eu estou totalmente cansado depois do exame. – also correct; adds a slight emphasis on eu (“I am totally tired today…”).

In everyday European Portuguese, you will very often hear the version without eu unless there is a contrast or emphasis (e.g. Eu estou cansado, mas ela não.).

What is the difference between estou cansado and sou cansado?

Portuguese distinguishes between:

estar = temporary state, condition
ser = permanent or defining characteristic

So:

Estou cansado. = I am tired (now / today) – temporary.
Sou cansado. = I am a tired person (by nature) – sounds like “I’m generally a tired type of person / I get tired easily”, and is much less common.

In your sentence, because you are tired today after the exam, it must be estou cansado, not sou cansado.

Is totalmente the most natural word here? Could I say muito or completamente instead?

Totalmente is correct and understandable, but in everyday European Portuguese, people might more naturally say:

Hoje estou muito cansado depois do exame. – very tired (most common, neutral)
Hoje estou completamente cansado depois do exame. – completely exhausted (a bit stronger / more dramatic)

Some nuances:

muito cansado – very common, neutral.
completamente cansado – stronger emphasis; “totally worn out”.
totalmente cansado – also stronger, but can sound a bit more formal or “bookish” in some contexts.

So for natural speech in Portugal, muito cansado or completamente cansado would be the most typical choices.

Why is it depois do exame and not depois o exame or depois de o exame?

The basic structure is:

depois de
- noun / verb

When depois de comes before a masculine singular noun with a definite article, de + o contracts to do:

de + o exame → do exame

So:

❌ depois o exame – wrong (missing de).
❌ depois de o exame – grammatically possible but not used in normal speech; speakers always contract.
✅ depois do exame – correct and natural: after the exam.

This same contraction works with all definite articles:

de + o → do (masc. sing.)
de + a → da (fem. sing.)
de + os → dos (masc. plural)
de + as → das (fem. plural)

When do I use do, da, dos, das in general?

These are contractions of de + the definite article o / a / os / as:

do = de + o
- do exame – of/from the exam (masc. sing.)
da = de + a
- da prova – of/from the test (fem. sing.)
dos = de + os
- dos exames – of/from the exams (masc. plural)
das = de + as
- das provas – of/from the tests (fem. plural)

Portuguese almost always makes these contractions; saying de o exame sounds unnatural except in very special emphatic or poetic contexts.

Why does cansado end in -o? How would it change for a woman or for more than one person?

Cansado is an adjective, and adjectives in Portuguese agree with the noun or pronoun in gender and number:

Masculine singular: cansado
- Eu estou cansado. (man speaking)
Feminine singular: cansada
- Eu estou cansada. (woman speaking)
Masculine plural: cansados
- Nós estamos cansados. (group of men or mixed group)
Feminine plural: cansadas
- Nós estamos cansadas. (group of women)

So in your sentence, cansado assumes the speaker is male. A female speaker would say:

Hoje estou totalmente cansada depois do exame.

Can I move hoje to other positions, like Eu hoje estou… or Estou totalmente cansado hoje…?

Yes. Hoje is quite flexible in word order, and all of these are grammatically correct:

Hoje estou totalmente cansado depois do exame.
Hoje eu estou totalmente cansado depois do exame.
Eu hoje estou totalmente cansado depois do exame.
Estou totalmente cansado hoje depois do exame.
Estou totalmente cansado depois do exame hoje.

Differences are mostly about rhythm and emphasis:

Starting with Hoje (your original) sounds very natural: focus on “today”.
Eu hoje estou… puts a bit more emphasis on eu (“I, today, am really tired…”).

In normal conversation, Hoje estou totalmente cansado… is probably the most typical.

Should there be a comma after Hoje in writing: Hoje, eu estou…?

Style guides in European Portuguese often recommend a comma after many sentence-initial adverbs like hoje, ontem, amanhã:

Hoje, estou totalmente cansado depois do exame.

However:

In informal writing (messages, emails), many native speakers omit this comma.
In speech, there is usually a tiny pause after Hoje, which corresponds to the comma.

So:

Hoje, estou totalmente cansado… – slightly more formal / careful writing.
Hoje estou totalmente cansado… – very common in everyday writing.

Both are accepted; you won’t be wrong without the comma.

How do I pronounce this sentence in European Portuguese?

Approximate European Portuguese pronunciation (not Brazilian):

Hoje – [ˈo.ʒɨ] – roughly OH-zh(ɨ) (zh as in French jour; final e is a very short, closed sound)
eu – [ew] – roughly ehw
estou – [ɨʃˈto] or [ʃˈto] – roughly (ɨ)shtoh (often sounds like “shtoh”)
totalmente – [totɐlˈmẽtɨ] – roughly to-tahl-MEN-ti
cansado – [kɐ̃ˈsaðu] – roughly kun-SAH-doo
depois – [dɨˈpojʃ] – roughly d(ɨ)-POYSH
do – [du] – roughly doo
exame – [ɨˈzɐ.mɨ] – roughly i-ZUH-m(ɨ)

Said smoothly:

Hoje eu estou totalmente cansado depois do exame.
[ˈo.ʒɨ ew ɨʃˈto totɐlˈmẽtɨ kɐ̃ˈsaðu dɨˈpojʃ du ɨˈzɐ.mɨ]

Is exame the usual word for “exam” in Portugal? How is it used compared to prova?

In Portugal:

exame = exam in a more formal or official sense, especially:
- exames nacionais – national exams
- exame final – final exam
prova can also mean a test or exam, but it is more general: it can be any kind of test, trial, competition round, etc.

In Brazil:

prova is very commonly used for school tests.
exame is used too, but often has a more specific sense (e.g. medical exam: exame de sangue = blood test).

Your sentence depois do exame is perfectly natural in European Portuguese if you mean an official or important exam.

Could I also say após o exame instead of depois do exame? Is there a difference?

Yes, you can, but there is a nuance:

depois do exame – most common, everyday, neutral.
após o exame – a bit more formal or literary.

Examples:

Hoje estou totalmente cansado depois do exame. – very natural in speech.
Hoje estou totalmente cansado após o exame. – correct, but sounds more like written narrative, news, or formal text.

Meaning-wise, both are basically after the exam.

Why is the verb in the present tense (estou) if the exam already finished?

Portuguese works just like English here: you are describing your current state (now), which is a result of a past event (the exam).

Hoje estou totalmente cansado depois do exame.
= Today I am totally tired after the exam.

The logic is:

Past event: o exame (the exam already happened).
Present state: estou cansado (I am tired now because of that).

You could mention the exam in the past more explicitly if you needed to, but in normal conversation the present estou is exactly what you want.