It’d be easy in Nagoya city if you speak Japanese. You want to run a google search for: “名古屋市 名刺 即日印刷” or ask your hotel front desk. The main problem is that most print shops won’t speak English and they all seem to require a few hours. You might want to order ahead and arrange to have them delivered to your hotel instead.
As for the business card template, I’d prepare it as a PDF or high resolution GIF or BMP. I’ve found print shops don’t always have the latest copy of Illustrator and fonts can vary. So embedding fonts into the PDF or having it as an image file is the way to go. Note that Japanese meishi are 90x55mm.
There are also meishi vending machines which have a touch panel interface and an internal printer but they aren’t easy to use if you don’t know Japanese.
I didn’t locate any stores with meishi printers in Chubu Airport. There might be a meishi vending machine but I can’t locate one by google.
Also, note that many of the big print shops on the internet can fedex you business cards anywhere in the world. They might be easier to work with.