Hi, if I back up to my mac using a specific ID, can I restored a new phone from that back up with a new ID? I'm going from an iPhone 4 to iPhone 5

Not sure, only the same name could restore from the same backup. But you could try to change the id later after restored.

You are getting new apple id? care to say why? or you just want to give the phone a different name? you can restore from the backup but the apps can only be updated with the old id.