I've tried running, the timing is very tight. I recently saw a video of someone doing it, and they used the slide move. The only difference is; if you want to get into the secret area instead of sliding outside the castle, delay your slide by 1/2 a second.
If you slide into the door right as its closing you'll get caught in screen flicker, just keep holding right and you'll appear inside the secret area. Also, don't do a slide-jump, just a regular slide.