Every ward in Tokyo will offer very cheap Japanese lessons led by volunteers. Many are free talk sessions, while others use a textbook. For a beginner it is best to find a structured class. Although I live on the opposite side of Tokyo, I found the language classes in Minato City to be the best. I attended “Sakura Kai” and they were excellent. Their website is available in English and Japanese, and they only charge 2000 yen per month, for twice a week two hour lessons.
They have three levels – Beginner 1, Beginner 2 and Intermediate. Every lesson involved learning first Hiragana, then Katakana, then eventually kanji, plus new vocabulary and a grammar point. Group work was often involved, and you are expected to learn the vocabulary mostly as homework. They assign homework every lesson and will check it for you.They also hold some cultural events like calligraphy or cooking classes.
The teachers were mostly retired Japanese locals, but they did a good job of organising the lessons, it was very clear and easy to follow the schedule. In the intermediate class things became more difficult as the teachers stopped writing Hiragana above the kanji. These classes will get you from zero Japanese to about N3 level (JLPT).
The classes run from September – June, see their website for dates.