“Punctually one hour” is non-standard; it has an adverb improperly followed by a noun phrase. “Punctually in one hour” and “one hour, punctually” are acceptable alternatives in some cases. For example, “They collect the papers punctually, after an hour goes by” or “They deliver the photos in one hour, punctually”.
The usual sense of punctually is “In a punctual manner; on time”. Note, punctually also has an archaic sense, “precisely; exactly; minutely”, by which it could substitute for precisely or exactly. And conversely, precisely or exactly can be substituted for punctually in the examples above with little change in meaning.
The adverb minutely, on the other hand, cannot so substitute, due to its senses “With attention to tiny details” and “On a minute [tiny] scale”. The adjective minutely also cannot substitute; it means “Happening every minute; continuing; unceasing”.