Translation:Al-Azhar is a very famous Islamic university.
I don't think so. It sounds a bit messy in English that way. Also, what comes after "is" is the point of identification (predicative) which the sentence is trying to tell. Thus in the sentence above, the identification is for Al-Azhar to be a very famous Islamic university, while in your sentence, Al-Azhar which is an Islamic university is identified as very famous only.
"a smart boy is hungry" is somewhat not directly translated into Arabic, but it depends highly on the context. Maybe we can say something like هناك ولد ذكي وهو جائع (there is a smart boy and he is hungry). This is because Nominal sentences in Arabic (sentences starting with a noun) typically have the subject defined (with AL) and the predicative of the sentence would typically be indefinite. So, starting with an indefinite noun is unlikely. Translating by the context and meaning is the key here (and in every situation mostly between any 2 languages).
"A smart hungry boy" would be ولد جائع ذكي or we can add "and" between adjectives as well: ولد جائع وذكي. Notice that this is not a complete sentence, as the adjectives here are both attributive (describing the noun) and NOT any of them is predicative (telling the status of the noun); i.e. there is no (to be) verb to connect the sentence in English to make a full sentence. Likewise in Arabic, the adjectives follow and mimic the noun in its attributes (gender, number, definition).