The characteristic Worcestershire-flavored sauce you are looking for goes by several names: okonomi, tonkatsu or yakisoba sauce. You will also need nori and pickled ginger to get the flavor just right.
I was looking at yakisoba, and god has mercy on me, I found this:
Not sure about using the sauce you saw on Amazon. But whenever i need a brown sauce in pan fried noodles I alternate between using a mixture of Soy sauce & hoison sauce or Soy sauce & Oyster sauce. Both sauces have distinct flavors and both have worked good once i mixed a bit of soy sauce in to cut the creamy pasty texture.