Very cheap udon noodle shop in west Shinjuku. It is served cafeteria-style.
For vegetarian/vegan, go to the first counter and order some udon without soup (eg "Shoyu" udon is just udon with grated radish and soy sauce).
Go to the next counter where, if you want, you can have tempura too (not vegan). There is eggplant, pumpkin, sweet potato, kakiage, etc for vegetarians. There are also onigiri (rice balls).
Pay at the next counter. Also put ginger and spring onions on your udon here.
Shoyu udon: Small 290yen. Large 390yen.
Tempura: 100yen each.
They also have stores in Akasaka, Kichijoji and Suidobashi.