Step 1: Notice that "hardly" makes the sentence negative in meaning.
Words like "hardly", "scarcely" and "rarely" act as hidden negatives, so "He hardly cares" behaves the same way as "He does not care much", even though there is no "not" written in it.
Step 2: Apply the question tag rule.
A negative statement always takes a positive question tag, and a positive statement always takes a negative tag. Since our main clause is negative in meaning, the tag must be positive.
Step 3: Find the correct auxiliary verb.
The main verb "cares" is simple present, third person singular, so its matching auxiliary is "does", not "will" or "won't", which belong to the future tense.
Step 4: Eliminate the wrong options.
Option B, "doesn't he", is a negative tag, but we need a positive one, so it is wrong. Option C, "will he", uses future tense, which does not match "cares". Option D, "won't he", is both future tense and negative, so it fails on two counts.
Final Answer:
The tag must be positive and in the simple present, which gives us "does he". \[ \boxed{\text{Option A: does he}} \]