First, I'd use 3/4" plywood and not particle board. There is virtually no support left on that particle board since it has a huge hole cut out of it. Yeah, you can build up support with the cross-pine pieces, but I'd worry about it cracking in half down the 2 sides on the left and right. At the very least I'd laminate it (put another layer on glued with plywood glue/gorilla glue) with another at least 1/2" to 3/4 layer of mdf, particle board or plywood to ensure strength.
Second, I'd HIGHLY suggest not installing it with the plastic still on it. Computer monitors are not meant to be in enclosed wood boxes, nor are they meant to be installed at a 80 degree angle. There will be no where for the heat to go other than behind the tube. Honestly I'd treat it as a fire hazzard myself.
I have a 13" Commodore monitor I've hacked for arcade RGB inputs, and have it rotated on it's side (but still facing foward). The plastic on that thing gets very hot and you can smell it after it's been on 1 hour or so, because it's not venting properly that way.
You'd be much better off removing the case, mounting the tube directly to the board (once you make it stronger), and then building a shelf under it to hold the electronic board with the components facing upward. There would then be enough open space for the heat to flow away without building up in a plastic box that also is sealed in a wood box.