Three of these are grammatically acceptable, but only #4 is a likely "natural" construction. The others all sound awkward, or even somehow twisted.
- 1) That I completed the drawing took a day.
This sounds as if the FACT that you completed it "took" a whole day. But the fact doesn't occupy time.
This construction would sound more natural in a sentence like: "That I completed the drawing [at all] was a miracle."
- 2) For me to complete the drawing took a day.
Hard to pinpoint what's "wrong", but reading this, I parse "For me to complete the drawing" as an adverbial clause rather than a noun phrase. I almost expect it to end something like this "...drawing, I first had to find my pastels." So after I have read to the end, I then have to re-parse the beginning as a noun phrase.
Anyway, it feels more natural as "For me, it took a day to complete the drawing." This would have the connotation "Other people may have completed theirs sooner, but for me it took a day.
- 3) Me completing the drawing took a day.
This needs "My completing" rather than "Me completing". Even so, it would be more natural to move "me" toward the end: "Completing the drawing took me a day."