Honestly, I think you're mostly on the losing side of the argument as far as artistic or cultural subjects are concerned, and considering the way this "hyper-relativism" is usually employed. When your interlocutor says "who are you to say, everyone is entitled to his opinion”, what he actually means is "I like this thing, I don't understand why I like it well enough to articulate the reasons to you, and I don't like that you don't like what I like so please shut up".

It's not a question of refuting the claim "everyone is entitled to his opinion", taken to mean "everyone knows what pleases him" in this context, because that interpretation is simply the truth. You have to reorient the discussion around what people should like which is different. Normative vs positive. You have to move the other person from a world where the purpose of consuming art/culture is to induce pleasure, to a world where one uses art/culture to cultivate taste. That's a lot easier said than done. I find it helps to be as specific as possible about what you like about e.g. the screenplay of Taxi Driver, because you will be more readily understood and you will find it easier to talk about objective quality.

I don't expect this to work all that often. When I was young, I enjoyed listening to hymns at church. When I was older, I noticed how mind-numbingly simple the melodies were. Presumably they are simple because they're intended to be sung by the whole congregation, which includes people who are ... not savants. People like that usually are not going to appreciate Mozart's Der Hölle Rache the way an opera aficionado would, though it can happen.