I have been glad for every stage I did, including times I didn’t get a job. Now I sometimes get to bring people in to stage, not as an application but on their request to experience the work, and they’re some of the most rewarding things I get to do.
The chef invited you because they want you there, they like the cut of your jib. Go in curious and enjoy it. Be honest about what you do and don’t know, any task you’re assigned they are obliged to teach you. The restaurant probably does some things differently from how you usually would, which means you will now know two (or more) ways to do that thing. By all means be respectful, but you don’t need to be overly deferential; a modern stage is reciprocal, you’re doing each other a favor. And you really are interviewing them, if you get bad vibes there are a lot of restaurants who will teach a cook who shows up to learn.
They may ask you to cook something of your choice, so think of a few things you might be able to make with ingredients they’re likely to have. Beyond the obvious you’ll have miso, soy, kombu, garlic, ginger, brown sugar, sake, and probably rice, panko, and tahini. I might do a pan fried noodle dish. Often they’ll give you a protein, usually a chicken breast, or might even ask you if you know how to cook a specific protein and give you that.
If you show up with a Fibrox everything will be fine, it’s still the best knife with a plastic handle. I doubt the chef expected you to buy a new knife, but if you want one anyway go for it. I can empathize, I still have the Zwilling paring knife I bought the day before a stage terrified of being caught unprepared. Surprisingly the chef didn’t rifle through my knives. I don’t use the paring knife much.
I will say at $45-50 the Fibrox chef knife has more than doubled in price in the past ten years. The Mac Chef series has been my recommendation for a long time. They’re more, $70 for a 7.25" (180mm) and $80 for the 8.5" (210mm), but for many cooks it’s worth the money. There are several other options (Tojiro, Fujiwara Kanefusa, maybe a local store if you live in the right metro) but I promised myself I would not badger people about knives anymore.
The shoes, though, buy. Even if they’re Walmart non-slip shoes, or Crocs, nobody wants you falling.
Good luck! It sounds fun! I have never staged at a ramen place, and I doubt I’d take a job at one, but I’d be interested in doing a day or five now.
I will also suggest, I know unsolicited and likely unnecessary, you keep your tech resume updated. A lot of cooks spend years and a lot of their disposable income to get the sorts of degrees and certifications you likely have. Maybe cooking full time works out, maybe a part time gig at the ramen place or somewhere else scratches the itch, maybe the hobby remains an untainted hobby. Nothing wrong with keeping your options open.