How to save data to database without page refresh using jQuery in ASP.Net

Refer

Submit form without page refresh (refreshing) using jQuery AJAX in ASP.Net

Or you can use Update Panel to Partial Postback.

This worked

But if the method is invoked with an error, what should be?

As you can see in the article which you have mentioned in your question.

Calling ASP.Net WebMethod using jQuery AJAX

You can gerenerate the Responce message and show this on

 function OnSuccess(response) {
    alert(response.d);
}

You can add  ADO.Net success or error messages in some string variable and return it. It will be shown in the OnSuccess javasscript function.

Refer

Display error messages and details of error occurred during jQuery AJAX Call