Kyoto:
Gion Nanba is a bit more expensive than your specified budget, but an excellent restaurant. Approximately JPY 20,000 per person.
Gion Suetomo is also fantastic, although a bit more expensive still. Both of these restaurants are well worth it.
Tokyo:
Yakitori Toriyoshi (Akasaka Mitsuke is closest to Shinjuku)
Tempura Mikawa (Roppongi or Monzen Nakacho)
Imahan in Odakyu Daiichi life building (adjacent to Hyatt Regency Tokyo) for Sukiyaki/Shabu Shabu etc...
Kokugyu in Yoyogi is a good yakiniku restaurant that offers good value (usually around JPY 10,000 per person)