there is no good way to fix it properly without replace that whole side of the cabinet (at least back to what it was). The problem is it is most likely a melamine and not actually a laminate from what i can tell. The fake wood veneer is thermally fused right to the particleboard and has virtually no thickness to it. You will be very hard pressed to match the exterior in a laminate (and even if you do, a sheet of laminate isnt particularly cheap.) Then you uave the inside to worry about, and i assume they knocked out some of your adjustable shelf pin holes, and recovering the whole inside face would cover those up.
However you could get creative with it if it seems otherwise structurally okay. If it were my rental, I would strongly consider cutting the whole out square, and filling it with some uv prefinished maple or some other similar looking 1/2 inch ply with a clear cost from your local building supply store. Then you could get a piece of 1/4 " thick mdf and some cheap trim and some chalkboard paint, and create a framed chalkboard that completely covers the hole from the outside.
EDIT: I am an engineering manager at a cabinet and Millwork shop by day.
