I read that English has the following degree of comparison: positive, comparative, superlative (with definite article) and elative (with indefinite article). I'm wondering, whether sentences like "it is a most brave action" can be used in modern speech or it will be said "it is the most brave action"?
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The use of the indefinite article with most is a time-honored way of making a symonym of very.
See examples here, or in the dictionary of your choice.
Another example that occurs to me now: the Simon & Garfunkel song, A Most Peculiar Man.
In English, there isn't as well-defined an elative degree as in some other languages. Nor is there a well-defined difference between absolute and relative superlatives as there are in some others.
We do though use superlatives in an absolute sense with the indefinite article, much as per your example, and this is indeed sometimes referred to as elative but that label may cause more trouble than it's worth. (It doesn't exactly match the elative degree in Latin, and elative has two other uses in describing features of other languages that English doesn't share in the slightest).
Because superlatives are generally taught in their relative rôle (generally with the definitive article), their importance in an absolute rôle is often understated or even ignored.
However, they are indeed used. The form "a most important day" is relatively rare, but only relatively; it is certainly not strange. The form "I am most happy to hear this" is quite common.
While it doesn't answer the querent's original question about modern use, the comments have suggested that this isn't grammatical. Now, not everyone may agree but to my mind if Shakespeare said it, then it's grammatical:
I also tend to take the same attitude to the grammar of the KJV. I can't find any "a most" in the text and to search for other indefinite article + superlative combinations would be more involved, but "a most" does occur several times in the translators notes.