You'll have to take a few minutes walk from Ginza, but the place called "Oniyamma" in Shimbashi is highly recommended. The place is standing style so it's not a place to sit and relax, but you can try the delicious udon casually!
"Sato Yosuke" is the place to go!
They offer traditional Akita style udon called Inaniwa Udon, which the noodles are very thin comparing to Sanuki udon.
At "Sato Yosuke", you can try other small dishes of Japanese food, too. Like Yakitori and Tempura.
Here is the official website : http://www.sato-yoske.co.jp/shop/ginza-en.html