Four in ten have an English menu
Of 567 restaurants, 224 have an English menu — 40%. The other six in ten run in Japanese only.
But that 40% is not spread evenly. By genre it splits hard: either nearly every shop has one, or almost none do.
The places tourists already know
Izakaya: 20 of 20. Yakitori: 20 of 20. Yakiniku: 22 of 24 (92%). Kushikatsu: 16 of 20 (80%). Sushi: 16 of 21 (76%).
These are the genres the world already associates with eating in Japan, and the shops are set up expecting foreign customers. You will rarely struggle here.
The Nagoya specialities do not
Turn to what the city is actually famous for and the numbers invert. Kishimen: 0 of 14. Ankake spaghetti: 0 of 15. Miso-nikomi udon: 2 of 23 (9%). Ramen: 3 of 25 (12%). Miso katsu: 11 of 30 (37%).
The reason is simple — these are what local people eat on ordinary days. They never developed inside a tourist economy, so the shops still run in Japanese. The better known the dish is as Nagoya's, the higher the language barrier.
Cash-only lands in the same column
166 shops (29%) take no cards, and the same skew appears. Ankake spaghetti: 14 of 15 are cash-only (93%). Ramen: 20 of 25 (80%). Kishimen: 8 of 14 (57%). Udon: 14 of 25 (56%). Miso katsu: 14 of 30 (47%). Kissaten: 14 of 33 (42%).
The shops without an English menu and the shops without a card reader are very nearly the same set. The older and more local the business, the more likely both walls stand at once.
How to make it a non-issue
First, carry cash. A few thousand yen in your pocket on a Nagoya-meshi day removes most of the problem outright. Convenience store ATMs take foreign cards around the clock.
Second, point. Every shop page here prints the shop name and the dish in Japanese — show the screen. It works better than attempting the pronunciation, and staff are used to it. Whether an English menu exists is also on each shop page, so you can check before you leave.
Which trade you want
You could eat only where both an English menu and a card reader exist. Three days of izakaya, yakitori and yakiniku would pass without a single difficulty in language or payment.
It would also mean coming to Nagoya and not eating Nagoya. Kishimen and ankake spaghetti have no English menu, but they arrive perfectly well when you point. The height of the wall has nothing to do with the value of the food. The numbers say prepare — they do not say avoid.