Yes, a high-speed Railjet train runs between these beautiful Austrian cities. It's a great option to choose for your journey, as it boasts not just fast travel times, but also provides everything you need for a comfortable ride and offers hourly departures!
No, unfortunately, taking a night train is not an option as Salzburg and Innsbruck are nestled within less than a 2-hour ride. But fast and comfortable Railjet trains offer hourly departures and boast amazingly fast travel times, so it's very easy to plan a trip so it would work with your travel plans.