Before you do anything, are you sure that the drawer/shelf thingy is going to be at a comfortable working height for a desk?

At cursory glance, what you want to accomplish is doable, but you'd have to use drawer slides that could withstand the weight of you leaning on it/equipment/etc. From there you'd simply add hinges to the shelf face. To operate it, you'd pull the shelf out, then hinge the face downward. My concern is that the drawer face is going to get in the way no matter what you do with it and the desk will look like poop without it when the shelf is in the closed position.

What is the part protruding at the top of the drawer? Is it part of the door or part of the face frame of the cabinet? If it's part of the drawer, then I would recommend "flipping" the drawer face over, so that is on the bottom. I would remove the protuberance and affix it to the shelf and affix the drawer face to the cabinet, then you simply pull the trim protuberance out with the shelf attached to it, maintaining the look of the cabinet while giving you unencumbered work space.

You would simply sandwich the edges of the pull out shelf between two pieces of wood to create a slide that will support both the top and bottom edge of the shelf. You'll also want to put a stop on the bottom of the shelf so that it can't be inadvertently pulled out of the cabinet. You'll need to leave 4" to 6" of the shelf inside the cabinet to combat the cantilever of the shelf OR install some pull out supports for the shelf, but you'll still need a stop on the shelf to keep it in the cabinet.