How to conceptually represent attributes (characteristics) of an object in logics?
For this discussion, I'll use classical first-order logic. I'll interpret your term "first-class citizens" to mean those objects which can be quantified over or passed as arguments to predicates and functions. The examples provided below are toy examples, purely for expository purposes. Have a look at the leading AI textbook which gives a much more extensive, proper introduction to this subject.

There are often many ways to formalize a given statement in logic. Let's consider your example 'An apple #1 has price 1$'. There are some naive ways that come to mind:

price(apple1)               this is a function which yields the price
Price(apple1,80)            this is a predicate

As for your question, "Is my representation of attributes by whole-part-relationships acceptable?", it at least raises eyebrows. Generally, whole-part relationships should reflect the notion that the part is part of the object structure itself, not some attribute of the object. So while has-part(apple,stem), has-part(apple,skin), has-part(apple,seed) seem reasonable, has-part(apple,price) does not.

Now for your main question, "how to represent attributes in the logics in which attributes are not first class citizens?" you can use either functions or predicates. Examples are given under the paragraph above describing some ways one can formalize a given statement in logic.