Learn English – Phrase Usage:- “Works like a Charm”, can it be used formally

(First off, it's much much much more natural to say "a friend of mine". I tried but failed to come up with a natural-sounding sentence that uses this phrase with a definite article.)

You can, grammatically speaking, use the "of mine" construction with pretty much any noun that you are capable of possessing.

That looks very much like a fork of mine.
We used a book of mine to settle the debate.

However, be aware that this construction always sounds a bit stilted, and it means something slightly different than the straightforward "my [noun]" form.

He drove a car of mine to the restaurant. [The car is one that I happen to own; I probably own others.]
He drove my car to the restaurant. [The car is probably my only vehicle, and it's the one I usually drive when I go someplace.]

If in doubt, stick to using "my [noun]" except in the set phrase "a friend of mine".

"At the moment" implies a short time, as, he was bust at that moment. There is an expectation of change relatively soon after.

moment n. 1. A brief, indefinite interval of time.

If the time in the past lasted longer, you can use other prepositions regarding time:

during, at the same time, at the time, in the meanwhile, for the time being, while, etc.

Since your study of English was not momentary, I would use a phrase that connoted a longer time interval.

In 2001 I watched a great documentary about Shakespeare. I was learning English at the time, and this was the reason that novel books attracted my attention...

I'm not exactly sure how you are trying to link novel books with Shakespeare. Shakespeare mostly wrote plays and sonnets. If you are saying you were attracted to his work, you could say exactly that:

In 2001 I watched a great documentary about Shakespeare. I was learning English at the time, and for this reason I was attracted to his work/for this reason his work attracted my attention...