I hope other, more knowledgeable people can help answer this question, but I will try to give you some information to start.
Since I don't know much about Islamic Dietary Laws, I took a quick look at wikipedia for some information, so this is what I found. I will quote wikipedia then follow up with some food suggestions. Also, I don't know how strict you are, so please read the following carefully!
From the sentence, "Dhabīḥah (ذَبِيْحَة) is a prescribed method of ritual animal slaughter; it does not apply to most aquatic animals..."
I assume that means that fish and any sea animals are safe to eat. If this is true, then sushi, takoyaki, and many kinds of okonomiyaki are safe for you to eat.
Sushi is made of vinegared rice and various toppings. Usually the toppings are raw fish, but other kinds such as vegetables and cooked egg are also popular. Many kinds of sushi have a little wasabi (a spice made from a root plant) and are eaten with a bit of soy sauce.
Takoyaki are small pieces of octopus cooked in a round dough ball and usually topped with sauces. I'm not a big fan and don't eat this often, so I can't remember the name of the sauce, sorry!
Okonomiyaki, especially in okonomiyaki style restaurants, can be literally anything included, but I think the most common kind is a mix of egg, cabbage, and a few other ingredients with mayo and a sweet sauce on top. If you are wanting to try, there are many vegetarian kinds (I think most of them are, actually), so just be careful to choose one without meat!
As for ramen, many kinds contain meat (usually pork I think) both as a topping and sometimes in the broth. So that may not be so safe depending on how strict you are. At the very least, you can order the ramen without the meat topping, and if you get miso ramen the broth may be okay. However I'd be careful.
When it comes to any other meat, unless it is in a Muslim restaurant, it will not be halal. Wikipedia says, "The animal must be slaughtered by a Muslim or by one of the People of the Book, generally speaking, a Christian or a Jew, while mentioning the name of God (Allah in Arabic)". Since the majority of the population is Buddhist or Shinto or both, this is obviously not going to be the norm throughout the country except at special halal restaurants. So I think it is safe to say anything else with meat is bad.
Since it seems there are no restrictions whatsoever on vegetables, checking this sites many forum questions about vegetarian foods will also be helpful for what you can and cannot eat, but I can add a few more foods that I believe are safe:
Pickled veggies, which come in many many kinds, are made with vegetables and vinegar.
Natto, a Japanese food many love to hate, is fermented soy beans, and is vegetarian.
Noodles such as udon or soba as well as ramen as long as you are careful about the broth!
There are many fried, baked, or grilled fish dishes as well.
Tofu, especially plain or with a little soysauce, is also a-okay. Be careful of the dish containing tofu such as Mabodofu though, this often contains meat.
Anyway, I apologize if any of this information is incomplete or unhelpful, but I hope you find some things you can eat and have a wonderful trip to Japan!
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