I'll play devil's advocate and recommend you keep the PC innards inside an actual PC case. Obviously if you want ultra arcade realism, taking the mobo and related parts out of the case and mounting them in the arcade cab is the best option. If you don't care about that sort of thing though, it's far better to leave the PC parts in a standard PC case. PC cases are designed for optimal airflow, and even though it's cramped inside, your PC will actually run cooler inside a case rather than exposed to the air inside a cab.
BUT, this is MAME were talking about, and it's not exactly taxing on a CPU(especially if you''re running a fast CPU), so it's not going to cause your computer to heat up that much.