Don't miss out on trying soba noodles at a highly dedicated soba restaurant - where the flour is freshly milled and the noodles rolled out and cut that day, a quintessential Japanese experience (and one that won't break the bank). There are a wide variety of soba noodles, some are light made from highly milled buckwheat flour, others are dark with a stronger flavour - then there are the many kinds of flavours that are incorporated into the noodles which include green tea, ume, yuzu, sesame & shiso leaves. Ramen is a result of the indigenous development of a Chinese import (wheat noodles) whereas soba noodles are an entirely Japanese invention dating back about 400 years.
Out of the suggestions on this web page:
http://metropolis.japantoday.com/tokyo/433/dining.asp , I can recommend
Toshian &
Kanda Yabu Soba. (My favourite is a restaurant in Yotsuya, but I can't access my notes for another month to tell you the name of it).
EDIT: Found it! Not Yotsuya but near Iidabashi station.
Shinano, lunchtime and evenings (closes 8:30pm) closed on Sunday. 2-10 Kagura zaka, Shinjuku ku.
And a great place to try several kinds of soba is
Hojinbo B1, Amano Building, 1-5-10 Nishi Shinbashi, Minato ku - also closed Sundays.
I ended up making one order and getting a serving of 'standard' soba, followed by a serving of ultramilled 'sarashina' soba flavoured with yuzu peel, and then sweet dark 'inaka' soba which has the strongest flavour of buckwheat. All served cold, mori-soba style. Nearest station is Uchisaiwaicho.
Sake and soba go WONDERFULLY together, and cold noodles on a hot day are a delight as well as a relief. (Morisoba - or zarusoba wich comes with nori)
When you're doing your research on which restaurants to visit, don't forget to check what food is in season when you're there and choose a restaurant or dish where this kind of speciality is best served. Some of the best restaurants in Japan/Tokyo only serve a very narrow range of options and are highly specialised.