> How can a person of type A claim any kind of universal or objective meaningfulness of anything?

For arguments for objective meaningfulness (or good), I'm partial to natural law theory, which is basically that what is good for a human being is what fulfills the nature of a human being. I don't have time to get into details here, but suffice to say that murder, adultery, lying, and stealing do not fulfill the nature of a human being, whereas justice, fortitude, prudence, and temperance are examples of things that do.

I recommend the following for arguments for natural law theory, by philosphers of an Aristotelian/Thomistic bent:

[Book] Aquinas, by Edward Feser. Chapter 5 is on Ethics.

[Article] The Metaphysical Foundations of Natural Law, by David Oderberg: https://docs.google.com/file/d/0B7SKlRTfkUiebnUxMUEtc1B4VTQ/...