One thing I do like about this plan is that they realize that to keep costs down, they'll have to construct the pizzeria dome from concrete consisting of lunar regolith. But that's about the only part of the plan I like, the rest is just fanciful, yet imaginative, chaff.
In fact, it's more likely a late Japanese April Fools joke, a suspicion that grows stronger after watching the awful introductory video on their website.
Currently, there are no solid plans to go back to the moon at all, and although political ideologies change all the time, I doubt we'll see any significant human presence on the moon for many decades (much to Neil Armstrong's frustration). Sure, Japan isn't NASA, and they have their own spaceflight program, but I'm pretty certain there is little talk of pizza at JAXA HQ. There's also the commercial space sector; SpaceX could be hired! Somehow, that seems unlikely.
Oh yes, and there's nobody on the moon to eat said pizzas. No people plus no plans to put people on the moon equals no point in putting a pizzeria on the moon.