I can see doing this with four hand tools. A bench plane, router plane, chisel, and saw. You'll also need a good way to hold the wood in place while you work on it.
The bench plane should be able to take fine shavings. Ideally a smoothing plane, but a jack plane or even a block plane should work fine. The saw should be a rip saw, preferably something with a thin plate, like a tenon saw. You could get away with a regular panel saw or a cheapo miter saw, but you'll need to increase your tolerances and do more work with the plane.
The steps are basically the same as shown in the video, with the chisel and router plane standing in for the router table.
1) Get your 2x4 down to the final width and length you want for the card holder. Find a section of 2x4 that has no knots, and use that. So you'll have a block of wood about 3 inches long, maybe 2.5 inches wide, and 1.5 inches thick.
2) Use the plane to flatten and smooth one wide face of the block.
3) Mark the three edges of the recess you want to make to hold the card.
4) Use the chisel to cut into the one mark that goes across the grain. Make sure you don't cut too deep. You can use a marker on the back of your chisel to mark the depth of cut.
5) Use the router plane to cut the recess to a uniform depth. You could do this with a chisel in theory, but a router plane is the right tool for the job. If you do use the chisel, use a mallet to make shallow cross-grain cuts, walking the chisel away from the cut line you made in step 4. After the recess has been roughed out, push the chisel straight in from the open side of the recess. Make sure the chisel is sharp and be careful not to blow out the closed side of the recess.
6) Use the saw to slice off the recess you have created. (basically the sides and back of the card holder). Leave extra thickness so you can correct saw mistakes with the plane.
7) Use the plane to clean up the back side of the piece you just made. It will be a little tricky to hold this thin piece in place while planing. Sandpaper might be a better bet for this step.
8) Plane the face of the block nice and smooth again. This surface will be an outer face of the card holder.
9) Use the saw to slice off the back of the card holder. The sawn surface will face to the inside of the card holder, so it doesn't have to be super-nice, but you should at least sand down the areas that will be glued.
10) Glue it together.