Learn English – “a right way” or “in a right way”

Yes, but ...

Saying "excuse him" is usually patronizing, implying that the other person is not responsible or polite enough to make his own apologies. If you're talking about a child, "Oh, please excuse my son, he doesn't know any better", well, we expect children to not be responsible. But if you say it about someone else, "Please excuse Fred", that's almost certainly insulting to Fred.

It may be, of course, that your intent is to say that Fred is irresponsible or rude, in which case "excuse him" would be totally appropriate.

The one example I can think of where it would not be insulting would be if the person could not be expected to know that something he did was inappropriate. Like, "Oh, please excuse Fred. He's still learning English, he didn't mean that statement the way it sounded."

Your question is quite the stylistic one.

Indeed, the usage of 'in' may be considered as weird (link). The thing is, 'do in' means 'do in a certain location.' Hence, I have a question: what do you mean when say 'do in a/the right way'? No doubt, you will be understood...but it is not preferred (my personal experience says so, but you can resist if you wish).

On the other hand,I am not sure if I did it the right way implies how you acted. You did it (how?) the right way. And, yes, it is idiomatic, plus lexically and grammatically correct. As proof, Forbes actively uses it.

EDIT: Despite a verb you use (do, deal, etc.) the situation is equivalent: deal the problem (how?) the right way (explanation above), and it is an idiomatic expression (example of usage).

Again, if taken under scrutiny, the preposition "in" is misused.