What do you intend to mean by these two phrases? I could see both being used

Could you quote some examples in current English?

"Canadian foodstuffs and raw materials were of first importance in maintaining the Western Allies."source - typically it is the phrase "of first importance" or "of second importance", rather than third or fourth, that is used.

"In the Bible, man's second disobedience may be taken as murder, Cain killing Abel." source. However this makes disobedience seem like a countable noun meaning "an incident of disobedience"

There is no established use of "disobedience" or "importance" as count nouns. We can't say three disobediences" or *"five importances". Count nouns are those that can combine with the cardinal numbers *one, two, three etc.

Just because you don't normally count a noun, that only means that you don't count it as an unmodified noun. While we don't say two disobediences, we do still say two instances of disobedience. So first disobedience can make sense in such a context.

In which case "instances" is a count noun, not "disobediences", which is simply the complement of the prep "of", not the head word.

Don't forget that John Milton was a figure in the 17th century. Just because he used "first disobedience" does not mean it is correct to do so in contemporary English. Do you think it is correct or not?

@Apollyon you are right, it seems like almost all the google results are for Bible-based meanings or are for Equestrian penalty points. I think the answer to your question may be "usually no, but yes in some cases depending on the word and intended meaning" You may have just chosen outliers as an example. If you are thinking of using these two words specifically, including a sentence may make this question more easily answerable

"First" is an ordinal number, not a cardinal one, so it has no relevance here.

@BillJ the OP specifically asks ordinal numbers and mentions "first" , "second" but does not mention "two" or "three". You were the one to bring those into this conversation.

Is it natural, in contemporary English, say "I was amazed by his first intelligence" or "I was shocked by his first rudeness"?

@Apollyon no, those two are not natural. I do not know what you mean by the first phrase, but the second phrasing is understandable (but sounds like its coming from a non-native speaker) and would probably be phrased as "I was shocked by his first display of rudeness". [At least to my Midwest-AmE ears]

@katatahito The question has been editted. The OP called them cardinal numbers in the first place. Also, it actually makes more sense to discuss about the admissibility of cardinal numerals in the NP structure.

I didn't use "cardinal numbers" in the first place.