Updating your AutoPay information should not impact your bill payment cycle. Your payments should go through on time, just as you expect. However, if you want to double-check that the change went through as you expect, you can do two things.
First, you can continue to check back in My CenturyLink and watch for the update to your AutoPay payment method.
Second, you can monitor your bill for the next billing cycle to ensure the payment went through on time. If it did not, you can make a one-time payment to ensure your account is up to date.