If you'd like one of the above times, let me know?
Regardless of whether this is meant to be spoken or written, there should be a period at the end (at least I believe so). This is because the words are not questions, they are instructions, statements.
Your statement can be separated into two parts:
[If you'd like one of the above times], [let me know].
Question words include: will, who, what, when, where, why, how, is, can, are, do. 'If' is not a question word, it indicates a condition, if A, then B. Your condition is, there is a selection of times, on the chance/condition that one of them suits you, then do B, let me know what that time is.
If you wanted to turn it into a question, you can say something like:
"If you'd like one of the above times, CAN you let me know [please, ASAP]" "Are there any times that suit you?"
Is this intended to be and written as a question?
If not, how and why would a question mark appear at the end?
The answers to the above two questions will determine when (and whether) a question mark is the proper punctuation at the end of a sentence.
Is it clear that your sentence needs a question mark at its end?
(The sentences in this answer hopefully illustrate the point, don't they?)