"He is going to arrive tomorrow night/ on the coming night" is good to describe the night of the following day. For example, today is Saturday. These phrases would refer to Sunday.
"He is going to arrive on the next night" refers to the night after a stated date at some time in the future. For example, you could say, "We're returning from Mexico on the 20th, and John is going to arrive on the next night." (That would be the 21st.)
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