At the moment, I am trying to resize the bounding box of the pixel art shown in the picture below. It is too big and I can't work with it like this. Is there any way to resize it in Illustrator without changing the dimensions of the graphic inside?
It looks to me like there is a pixel box around the image that is reflected in the size of the bounding box. Not know exactly what you goal is makes this answer somewhat random but here goes...I think you have three options: In illustrator put the image into a clipping mask and manipulate it from there OR open the image in photoshop and crop tight around your image. Or you could save it as a .png with a transparent background.
One more though. If you are in Illustrator CC you can use Isolation Mode. Hover your cursor over the box until it's outline is highlighted, then double click to direct select the box. The rest of the image should grey out so you know you are in IM. You can resize the bounding box as you like without affecting the image. Try this one first. Good luck!