So as a native speaker, I'll say both are correct in this sentence, and which you use would probably depend on the dialect of English you are speaking. I would say "care about me," personally, but "care for me" is absolutely OK, just wouldn't be what I, a Californian would typically say.
However, if I was to break down the actual problem with this sentence, just to my native speaker ears (and not to make you nervous about grammar, because honestly you're fine and this is just nitpicking) either nix the "the" you have before overprotective or nix the comma. Right now you have a "the" which indicates a noun, but overprotective is an adjective.
Always overprotective, older brothers, but secretly I loved that they cared so much (about/for) me.
or
Always the overprotective older brothers, but I secretly loved that they cared so much about/for me.
Thing is, this sentence is still a fragment becuase the phrase "always the olderprotective older brothers" is cut off by the "but." "but" kind of starts a new phrase here, and I don't know what this is called in grammar terms anymore, but either finish the phrase that Always makes before you get to but, or find a way to rephrase it that nixes the "but."
so, for an example
They were always the overprotective older brothers, but I secretly loved that they cared so much about/for me.
Thing is, you can leave it a fragment if this is in a conversation, or if you want your writing to sound more conversational and less formal, because we tend to speak in fragments a lot of the time, and the sentence made sense before. But as a sentence standing alone, "Always" is an adverb and it needs a verb to describe (in this case that verb is "were")
