Do I need a specific pot to cook Japanese-style rice?
In my experience, rice can be cooked in any pot. I cook quite a lot of Japanese style dishes, and as far as I can tell the rice is cooked in the same way as any other.

For myself, I've done it in good "induction-able" steel pots, as well as huge (navy galley) aluminium pots, and a couple of low-quality stainless steel things too. They're all fine.

Normally, as soon as the pot is at/near boiling, turn the heat down to a low setting and leave it in the pot until it's ready. Then take it off the heat, but leave it covered (don't peek) for another five minutes.
Generally, once you've added the water, you put the lid on. You only start the timer when it's boiled and you turn it down, though. That's probably why a glass lid is better, so you can see the boiling. It's not critical to the process, though.