Einsiedeln is easy to reach from Zurich without a car. The Südostbahn (SOB) runs direct trains from Zurich Hauptbahnhof roughly every hour; the journey takes about 55 minutes, changing at Biberbrugg. There is no direct route from Zurich without that change, but the connection is seamless and the total trip is simple.
From Zurich Airport, allow about 80 minutes including the connection at the main station. If you are coming from Lucerne or Zug, the route goes via Arth-Goldau and Biberbrugg.
There is no motorway to Einsiedeln — the road in winds through forested hills — which is part of why the town feels removed from the Swiss commuter belt despite being under an hour from the city.