How do I style a navigation block menu?
You can style it by writing css that target the elements of the navigation menu. If you use the inspector in your browser you should be able to figure out what classes the menu is wrapped in so you can properly style it. 

If you are using a custom theme you can start by adding a css file to a css folder. In this case I named mine style.css.

/path/to/custom/theme/css/style.css

.menu li{
    background: blue;
}


Then I came up with a name for my css library in mytheme.info.yml

name: MyTheme
description: My custom Theme
type: theme
core: 8.x
libraries:
  - mytheme/global-css

regions:
  header: 'Header'
  content: 'Main Content'
  footer: 'Footer'


Then I made sure that the library included our style.css file. 
In mytheme.libraries.yml I had

global-css:
  css:
      css/style.css: {}