To come in handy means that something becomes useful at some point. For instance, an umbrella may come in handy on a hike if it begins to rain, or a spare tire may come in handy if one of your tires deflates.
"Turn out" means occur, happen, or come about. If it turns out that your umbrella is useful, then it came about that your umbrella was useful or it happened to be useful. I don't speak Japanese but I think 起こる is close to what it means.