The connection wouldn’t be offered unless it meets the requirements for minimum connecting time. That said, an hour is plenty of time at some airports, like SIN, HKG or in the USA for domestic… but might not be at others!
If you miss the connection the airlines will be responsible for getting you to your destination. Travel insurance will cover delay for hotels and meals if not provided by the airline. Not all policies will provide a replacement flight though (which in any event will be covered by the airlines involved).
I would monitor on the day of departure. If the flight is delayed, ask to be switched to a non-stop SIN-ICN. on the day of travel agents at the airport have a lot of flexibility to get you to where you need to go.