Unfortunately there is no way to do this automatically. Your online account providers will have different methods and policies for updating the email on file.
Usually this will require verifying the account password and in some cases responding to a confirmation at the previous address. Because of this, you will not be able to use an automated service to update the information.
You can make a list of the most important accounts (your bank, bill providers, Google, social media accounts, etc) and go through and update them manually.
Next, go through your old email account and see what other sites you may need to change this information on.
Finally, ensure you keep your old email active during the transition and for some time after. This will help ensure you have transferred the accounts you need, and that you will be able to respond to verification emails if required.