This has nothing to do with “presque”. You can say:

• “j’ai de l’appréhension” (I’m afraid for no specific reason): “j’ai de l’appréhension avant mes examens”
• “j’ai une appréhension” (I’m afraid of something specific): “j’ai une appréhension, c’est d’oublier mon stylo le jour de l’examen”.

It’s a question of comparative lingusitics and I think could be asked on linguistics. The question would not be asked by a native speaker of a language that uses determiners in a way similar to French.

In English or in German in that sentence the noun would have been used without a determiner (apprehension / Besorgnis) but in French, a determiner is always required with an abstract noun even if it is unspecified. That’s the rule, so no article would not have been grammatical.

Here the partitive article is used because the type of apprehension is unspecified, as you say we could use “some” in English.

If the feeling was specified, we would have a definite or indefinite article.

Specified by a noun phrase:

Tout le reste du monde, les chaînes de leur vie, les tristesses du passé, l’appréhension de l’avenir, l’orage qui s’amassait en eux, tout avait disparu. (R. Rolland, Jean-Christophe)

Specified by an adjective:

J’avais une certaine appréhension avant mon premier jour de travail.

The end of your question shifts to concrete nouns, more precisely about materials. It’s a different question altogether, but still different from English.

-C’est quoi ton pantalon ?
-C’est du coton.

Partitive article. No article in English & German (it’s cotton/ es ist Baumwolle)

But with a material French can also use the preposition “en” and no article.

Ce pantalon est en coton.

(made of cotton / aus Baumwolle)