How to fit a background sprite with 4:3 aspect ratio in a 16:9 screen by simply making the extra content of the sprite from the bottom go outside the camera?

Click on this sprite in Project panel and then in inspector click "Sprite editor", there you can cut your Sprite.

Assuming the game is 2D,

1. In the hierarchy, right-click your MainCamera and select [ 2D Object > Sprite ] to create a new GameObject with a SpriteRenderer component.

2. Assign your Sprite in the SpriteRenderer.

3. Add the SpriteBackground script below. [ Add Component > New Script > name it "SpriteBackground" > Create and Add ]

SpriteBackground.cs

 using UnityEngine;
 
 [RequireComponent(typeof(SpriteRenderer))]
 public class SpriteBackground : MonoBehaviour
 {
     // When the game starts,
     void Awake()
     {
         // get references
         SpriteRenderer sr = GetComponent<SpriteRenderer>();
         Camera cam = Camera.main;
 
         // check sprite is not null
         if (sr.sprite == null)
             Debug.LogError("Sprite is null.");
         else
         {
             // if screen is portrait, there will be a gap at the bottom.
             if (cam.aspect < 1f)
                 Debug.LogError("Portrait screen means SpriteBackground has a gap at the bottom!");
 
             // get sprite size
             Vector2 spriteSize = sr.sprite.bounds.size;
 
             // get orthographic camera height/width in world units.
             float height = cam.orthographicSize;
             float width = height * cam.aspect;
 
             // calculate the required scale
             float size = width * 2f / spriteSize.x;
             transform.localScale = new Vector3(size, size, 1f);
 
             // calculate the position so that the top of
             // the sprite touches the top of the camera view.
             const float zPos = 100f; // distance from the camera
             transform.localPosition = new Vector3(0f, height - (size / 2 * spriteSize.y), zPos);
         }
     }
 }