The first sentence is indeed idiomatic, as you say, and is far, far more likely to be said or heard.
I'll go to it on Monday.
Also, just as a note, it's at least somewhat more common to use I'll instead of I will in most contexts, certainly in verbal speech, but also often in formal writing. The main reason we'd use I will is to emphasize will.
Sometimes people will use I will when they wish to create a very formal effect. And sometimes people will just say I will because it's an acceptable choice--perhaps a native speaker will be unaware as to why they made one choice versus the other.
I'll/I will go to it in the next Monday
is trickier to evaluate.
It's certainly far less common than the first sentence. Whether or not it's grammatical depends on the definition of grammar that we have in mind.
Many people would judge it ungrammatical if asked. However, many of those same people would probably not notice it if they heard it, and would understand it to mean the same as the first sentence.
Most modern linguists would say that grammar describes they ways that users of a language acceptably use the language. Therefore, careful grammarians would be hesitant to define it as non-standard or ungrammatical. The authors of The Cambridge Grammar of the English Language, for example, would likely designate it as acceptable to some speakers of Standard English, but not to others.
If we are very careful, we can see the first sentence as having two different but overlapping meanings:
A) I'll go to the event that will take place on Monday.
B) On Monday, I'll depart to attend the event (that we're discussing).
For example, if people were talking about an event that takes place on Tuesday, and they were discussing when they're planning to leave in order to get there, there's a somewhat higher chance that the second option would be used, although on Monday would still be much more likely, than in Monday and this also pertains if next is used.
Note that next day-of-week, week, month, and year is surprisingly complex. It could mean this coming Monday, and in some cases not this coming Monday, but the following Monday.
This ambiguity sometimes causes confusion or misunderstanding, even among native speakers.
In summary, go in the next Monday is unlikely at best, and unacceptable or ungrammatical to the sense of some or probably most.
In day-of-week, if it's said, would probably be used if the speaker had in mind within the time borders defining Monday. We might get a sense of this from a more common expression like it will happen sometime in the next day.
Especially in verbal speech, we tend to make choices that may appear odd or ungrammatical if we look at them in writing and ask if they're correct.
In summary, the second choice is odd at best, and ungrammatical to some or most speakers of English if they were asked to focus on it and evaluate it.
However, if someone said it, many people would accept it and understand it without noticing anything peculiar. A careful grammarian might be hesitant to call it ungrammatical for that reason.