If you plan on laminating the cab, you could use sheathing ply for all anyone will know. But if not, and as others have stated already, you'll spend many more times the cost of good materials in your time trying to make it look decent.
MDF doesn't really "suck up the paint" any more than raw plywood does. But, you need to use the right stuff as a base coat. KILZ primer works awesome for sealing MDF. If you want to throw a second coat on, virtually none of your finish coats of enamel will make through to the board underneath and your finish can be incredibly smooth if you do a light sand afterward. But using MDF requires proper building techniques developed with that material in mind. Plywood is much more forgiving on that front.
Commercial arcade machines used a number of materials. They were costs conscious as businesses tend to be, so they often used good plywood where strength was required and some pretty nasty particle board-like material to fill in where it didn't matter so much. Some later cabs went to mostly particle board construction to save money, but exposure to any amount of water with manufactured wood products is usually disastrous.
The material you don't see used very often is actually the best stuff for the job. But it's expensive. It 's called MDO (Medium Density Overlay) Plywood. If cost is no object, this is the stuff to use. It marries the best properties of MDF (super smooth, grain-free surface) and plywood (lighter-weight and better strength).
In the end, you are the one that will have to build it, look at it everyday and use it. So what you decide is good enough for your purposes will be good enough. But if it's really bad and you don't want to hear that sentiment from others, it's probably best not to post a bunch of pictures of it on the Internet when you are done

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RandyT