"Luckily for you" means the same thing as "fortunately for you." This is a fragment, a portion of a longer sentence, such as, "Fortunately for you, I happen to have an extra umbrella in my car." (Context: a sudden downpour, no parking space close to the theater.)
However, "lucky for you" is a bit different. A slightly longer, clearer version would be "That was lucky for you!" Or "How lucky for you!" Example:
I have taken my son for a blood test. His favorite tech is on duty, the one who always gets it on the first try, and distracts him with a joke. I say, on our way out, "Lucky for you that Miriam was on duty this morning!"
And this means it was lucky for you etc. Here, "lucky" is an adjective modifying "it".
If we wanted to use "luckily" instead, in this same context, it would be
Luckily for you, Miriam was on duty this morning.