How do I achieve negative padding in CSS?

It completly depends on the font you use, but in your particular case height: 34pxand line-height: 34px; do what you want:

h1 {
  background-color: #375E97;
  color: #FFFFFF;
  
  font-size: 50px;
  font-family: 'Righteous', cursive;
  text-transform: uppercase;
  text-align: center;
  
  padding: 0px;
  margin: 0 auto;
  
  height: 34px;
  line-height: 34px;
}
<!DOCTYPE html>
<html>

<head>
  <link href="https://fonts.googleapis.com/css?family=Righteous|Roboto:400,700,400i" rel="stylesheet">
</head>

<body>
  <h1>Lorem Ipsun Al</h1>
</body>

</html>

The one thing you didn't try is adjusting the line-height.

10.8 Line height calculations: the line-height and vertical-align properties

On a block container element whose content is composed of inline-level elements, line-height specifies the minimal height of line boxes within the element.

h1 {
    background-color: #375E97;
    color: #FFFFFF;
    font-size: 50px;
    font-family: 'Righteous', cursive;
    text-transform: uppercase;
    text-align: center;
    padding: 0;
    margin: 0 auto;
    line-height: .6;
}
<h1>Lorem Ipsun Al</h1>