But you're using a TV in it, right? Or do you have an arcade monitor? Either way, it's doubtful it will care about the line frequency difference, so a simple step-down transformer is fine.
You will need to get a 220V smart strip or similar. I'm sure they sell them for the European market. You should not attempt to use a 120V power strip of any form on 220V (e.g. using a plug adapter) as the circuit breaker may not be rated for the higher voltage, among other things.
Your other option, perhaps easier, would be to just run the whole cabinet off a step-down transformer. You may need a somewhat sizable transformer, but they're readily available.