How to get the Gameobject a script is attached to

I want to have a script attached to a gameobject and rotate the gameobject with it.

I want to initiate the gameobject from a different script (while the instance rotate automatically)

what is the easiest way to get the gameobject the script is attached to (from inside the rotating script)? (c#)

Literally gameObject. You want transform for rotation and stuff.

public class Rotate : MonoBehaviour
{
    private void Update()
    {
        gameObject.name = "Been up for " + Time.realtimeSinceStartup;
        transform.position += Vector3.up;
    }
}

That would be the easiest way, consider watching the nice basic tutorials for Unity.