What does "fancy asking me that!" mean? I reckon it's something like "how dare you ask me something like this" in a sarcastic way. Is it?
Peter has hit around the issue, but not really on the mark. In fact, fancy is a fancy (and somewhat archaic) synonym for imagine:
2. To imagine or suppose: "I fancy she is an exceedingly proud woman" (Jane Austen).
Your quote is an expression of wonderment mixed with indignation: "Imagine asking me that!"
Fancy asking me that!
is usually said in a surprised voice. It could mean
Unbelieveable asking me that!
Incredible asking me that!
and is more light hearted, but would not usually mean something as confrontational as
How dare you, asking me that!
Whether it is sarcastic or not would depend on additional context.