That material will not hold up to arcade abuse. The heavy handed players will
destroy that material in seconds.. and the monitor will crash and shatter.
Not only that.. but the challenge makes little sense.
If you do not have tools.. how can you drill button holes?
Most people who want the arcade experience will just try to build a
control panel that sits on a desk. Its cheaper, and easier.
Home Depot will cut 8*4 MDF sheets to whatever size pieces you
specify. My cavalier fits almost 3ft wide pieces, which are well fat enough for
most things Ive needed to build. My rear seat folds down, so that is the advantage I have with that car - is that I can load like 8 to 13ft long pieces in the
car : ) If you cant do that, Home depot also has a rental truck. Or, try a friends truck or car.
If you are lazy... dont paint it

But, MDF being so smooth may not need much effort in painting anyway. If its a little bumpy cause you dont want to sand the paint down.. who cares... If its black, it hardly matters.
If you are Insistent on the cheapie bookshelf, just make a pedestal cab. (no monitor. Just place it next to the TV/monitor) Use double the wood (2 pieces next to each other) for extra strenght.
Only problem is.. cutting. When you try to cut that cheap bookcase crap.. (esp without a good tablesaw) it may break appart in nasty ways. Its nice laminate will be ripped too.. and how will you cover the jagged/chipped/frayed cut ends?
A more interesting challenge might be making a cardboard cabinet. Maybe by soaking the carboard in glue/water, layering them, and waiting for them to dry. Not sure if it will work or how strong it would be.