I had this same problem with my new iPhone 5S. If you DO NOT have access to your old email address, this is what you should do... (It's quite a simple fix)
Go to appleid.apple.com and click "Manage Your Apple ID"
Login with your CURRENTLY ACTIVE Apple ID cridentials
Under "Apple ID and Primary Email Address" click "Edit"
Type in your old Apple ID
Click "Save Changes"
From there you have reset your Apple ID login to your old address.
On your iPhone/iPad you may now use this old Apple ID and your current password to sign in so that you may delete the account on your device and then reactivate with a new Apple ID. You will then want to go back to your appleid.apple.com account and change your primary email address back to your current email address. By doing it this way, you are not required to verify your email address, but are still able to accomplish what you would like. It worked perfectly for me, as I no longer have access to my old email address. I hope that this helps!!