If I have service with cricket, and I have a warranty. Can I get a new battery replacement at a Apple store with being covered by a warranty through cricket?

It depends upon who issued the warranty. If it is an Apple warranty you can (if it is still covered which is not certain with an old iPhone model like yours). If it is a Cricket warranty then if you goto an Apple Store they may charge you and you would have to see if Cricket would reimburse you. You might want to ask Cricket before doing anything.