The answer is: My neighbor asked my brother and ME to buy bread for her.
You can't say: "My neighbor asked I and my brother to buy bread for her", but you can: "My neighbor asked ME and my brother to buy bread for her"!
What is more: if you remove "my brother" there is only one posible answer:
"My neighbor asked ME to buy bread for her."
That is why the answer for this sentence is: ME.