In 1939 I believe East Night High School was located at Withrow High School on Madison Road in Hyde Park.
Withrow was called East High School before the current name was given.

I also believe (because a cursory search is not definitive) that East Night High School was located in 1919 in the old Woodward High School, in downtown Cincinnati.

Someone will likely be able to provide a more certain answer, or you could contact the Education or History Department at the Cincinnati Public Library.