"Moving" skybox closer, to emulate a smaller room? (VR 360 degree app)

this is not how a skybox work

A skybox will always be at the horizon no matter what. When your camera render the game, if any pixel of the screen would be empty because of far limit or because there is just nothing, it will render the skybox for that specific pixel depending only on the angle of the camera.

is your skybox the room itself? Maybe you should try to make a physical skybox, create a cube out of 6 planes or quads if you want it to act a little more like a skybox, add a parent to the planes and make sure it is perfectly in the center of that planes cube

Then add a script like that

 using UnityEngine;
 using System.Collections;
 
 public class ManualSkybox: MonoBehaviour {
 
     private Transform camPos;
     private Transform pos;
     // Use this for initialization
     void Start () {
         camPos = Camera.main.transform;
         pos = transform;
     }
 
     private void LateUpdate()
     {
         pos.position = camPos.position;
     }
     
 
 }

It is really important to use LateUpdate instead of update if you want your skybox to be smooth and not dependent of script execution order. (if the skybox update before you move the camera, the skybox will act weird)