"Punctilious" is built from the same root as "punctual" and "punctuation," both of which trace back to the Latin "punctum," meaning a point. A punctilious person pays attention to every small point or detail of conduct, much like a punctual person attends carefully to the exact point in time.
Using this root based reasoning, we look for an option that reflects extreme care over small points: meticulous fits perfectly, since it means being extremely careful about details, precise to the smallest point.
We can rule out the rest by testing whether they relate to "small points of detail" at all: prude relates to modesty, wasteful relates to poor use of resources, and timid relates to a lack of boldness. None connect to precision or exactness.
Let's summarize:
Spotting shared word roots between the question word and one of the options is a reliable shortcut in vocabulary questions.