O exame de aptidão física e mental muda segundo a categoria.

Because exame is a masculine noun in Portuguese, even though words ending in -e can be either masculine or feminine. So the correct article is o:

• o exame = the exam
• um exame = an exam

This is something you just have to learn with the noun.

Aptidão means fitness, ability, or aptitude, depending on context.

In exame de aptidão física e mental, it refers to whether someone is physically and mentally fit or qualified. In official or bureaucratic language, this often means a fitness evaluation rather than a school-style test.

They are singular because they describe aptidão, which is singular:

• aptidão física = physical fitness/physical aptitude
• aptidão mental = mental fitness/mental aptitude

Together, aptidão física e mental means physical and mental fitness.

Even though there are two adjectives, the noun they describe is still singular.

Because física agrees with aptidão, and aptidão is a feminine noun.

So:

• aptidão física
• capacidade física
• condição física

If the noun were masculine, then you would use físico:

• exame físico

Here, though, física is not describing exame. It is describing aptidão.

Yes. Mental is an adjective that works for both masculine and feminine singular nouns without changing form.

So you get:

• aptidão mental
• estado mental
• saúde mental

Unlike físico/física, mental stays the same in the singular.

Because the subject is o exame, which is singular.

• O exame muda = The exam changes
• Os exames mudam = The exams change

Even though the phrase includes several other words, the main subject is still just o exame.

Here segundo means according to, depending on, or based on.

So:

• muda segundo a categoria = changes according to the category

In this sentence, segundo is a preposition, not the ordinal number second.

Yes. Those are very natural alternatives.

• muda segundo a categoria
• muda de acordo com a categoria
• muda conforme a categoria

All of them mean roughly the same thing. In official or formal language, conforme and de acordo com are especially common.

Because categoria normally takes the definite article a in this kind of phrase.

So:

• segundo a categoria
• conforme a categoria
• de acordo com a categoria

Portuguese often uses articles more than English does. English might say according to category in some contexts, but Portuguese usually prefers according to the category structure here.

It is a very natural official-style expression, yes. It means something like physical and mental fitness exam/evaluation.

The structure is:

• exame de
  • noun

For example:

• exame de sangue = blood test
• exame de vista = eye exam
• exame de aptidão = aptitude/fitness exam

So this is a normal Portuguese pattern.

Because exame de aptidão is being used in a broad, classificatory sense: an exam of physical and mental fitness.

Using da would sound more specific, as if referring to the fitness of a particular person or a previously mentioned aptitude:

• exame de aptidão física e mental = a physical and mental fitness exam
• exame da aptidão física e mental do candidato = an examination of the candidate’s physical and mental fitness

So de without the article is the more standard wording here.

Yes, and in this sentence varies may even sound more natural in English.

• muda segundo a categoria = changes according to the category
• very natural English meaning: varies depending on the category

In Portuguese, mudar is common even when English might prefer vary.

The structure is:

• O exame de aptidão física e mental = subject
• muda = verb
• segundo a categoria = prepositional phrase explaining the condition or criterion

So the sentence follows a very normal Portuguese order:

Subject + Verb + Complement

That makes it fairly straightforward for an English speaker.

It would be:

Os exames de aptidão física e mental mudam segundo a categoria.

Changes:

• o → os
• exame → exames
• muda → mudam

The rest stays the same because aptidão is still singular inside the expression.