Is one more correct: "lucky for me" or "luckily for me"?
Both are correct. Lucky for me is simply shortened, which is grammatically fine.
Lucky for me is an elliptic form of It was lucky for me. E.g:
[It was] lucky for me [that] I saw the sniper before he saw me.
Luckily for me is a complete adverbial phrase, and not truncated:
Luckily for me, I saw the sniper before he saw me.
Lucky for me seems to have become more popular than Luckily for me since about 1980, according to Ngrams:
Luckily, as it's an adverb. Compare it to 'happily'. You wouldn't say:
Happy for me, I met my long-lost brother the other day.