There is a subject in that correct German answer:
Geht es deinem Bruder jetzt besser?
That es is the same kind of dummy subject as in the English phrase It is raining. Who is that it? Nothing. It's a dummy required by grammar.
German uses that phrase es geht + ‹Dat› + ‹Adj› specifically for telling the health of ‹Dat›. It's a set phrase. You have to use that one.
Ist dein Bruder besser jetzt?
That is grammatically correct but it doesn't inquire the health of the brother but his conduct. Consider:
Sie ist kalt. — She's a cold-hearted person.
Ihr ist kalt. — She's feeling cold.
This is because the dative object of a sentence tells who has to face the consequences of the action. “It” is cold, and she has to face the consequences. Hence dative, ihr.
While in a coupler phrase as A ist B, the subject A is the one who is assigned the property B. English isn't particularly keen on that difference. German is.
From that example you can also see that German clauses do not strictly require a subject. And there's even a an impersonal passive voice in German that requires the clause not to have a subject.
Another thing: Why do we place "besser" at the last position? Where can we place "jetzt" in a sentence?
That is because the default order of adverbials in German is time-cause-manner-place-direction. Jetzt is a time adverbial and besser is a manner adverbial.