Is it possible to use the article in the sentence below implying during this particular hour? or is it optional?
In the one hour
For something which I expect will happen after one hour, this particular hour, I would probably say
In one hour (from now)
Similarly, we can say "In two weeks from today" or "In about 45 minutes from when he arrives at home".
To start a phrase with the words "In the one hour", we can specify which hour we mean with a relative clause. This phrase does not describe a point in time when something will happen, it describes a duration when something will be going on.
In the one hour it takes me to travel to London, I will have finished that email
In the one hour between turning the laptop on and the battery running out, I managed to prepare for tomorrow. And in that same hour, my colleagues were all just sleeping!
You see? "In the one hour" doesn't really work on its own. But it can work if you specify which hour you mean. Or if you expect it to be clear which hour you mean, then you can use "In that (adjective) hour".