I think your selection will be just fine.
I've seen both 4ohm and 8ohm game designs. I'd go with an 8ohm speaker... Better to feed the signal into more resistance than it was designed, and get less current (and slightly quieter audio) than the alternative... which is to feed audio intended for an 8 ohm load into only 4 ohms and draw twice the current than the amp was designed for.
Keep your volume knobs turned away from full-max and you shouldn't approach the power rating of your speaker. You're not trying to pump out audio to compete with the rest of the noise in an arcade... you're just trying to hear if it's working right.