I'm going to wait for him one hour. That is how we say it where I live. "For" can be used, but not needed.
really? Where are you from? I would think a more common way to drop one of the "for" words would be a reordering -- "I will wait an hour for him."
same! I would say colloquially in the uk it is quite usual
I will wait for him one hour. I am going to wait for him. How long? One hour. The second for can be used but is not necessary.
I am going to wait for him sounds right to me.