You might want to post a photo.
If the "wood" veneer is some sort of PVC plastic, then many typical woodworking glues will not adhere well. Depending on how well it is bonded with the material underneath, you also run the risk of it separating from that with the extra mass on it.
But this seems to be one of those cases where the hassle of contact cement is warranted. Getting the positioning right will be the trick. I've used pins, or "hinges" of painter's tape to help with that.
Alternatively, a cheap and cheerful solution might be builder's construction adhesive, often sold in large tubes for use in caulking "guns". You have bit more time to position things before it sets. But once it sets, it is 100% permanent.
I see you are using 2-inch dimension lumber cut to length, so applying those in succession would be easier. You just have to make sure you keep the rows tight and square. Make sure you allow for wood expansion across the grain. That is, don't glue them super-tight against each-other. You aren't really making a panel, but "decking" a surface by adhering material to a substrate.
Dry-fit first! Nothing in life is perfectly square, especially factory furniture. You are going to shape some of the pieces to fit nicely and look square, and offer variable relief between pieces, or you'll find everything leaning toward one corner or another by the time you are at the opposite edge. I recommend using a few small shims between the pieces, as you would for decking. (Remove the shims once the pieces are in place, and move them to the new edge.)
A dry-fit may also allow you to mark the ends out on the bottom and top with painters tape, so you just assemble things in order to get the best look.
Ideally, the glued side will not have any finish on it, but for contact cement or construction adhesive this is less of a concern.
Surface prep will be up to the adhesive you use, so read the label.