Spotify finally capitulated and added support for Chromecast streaming recently, but there were a number of caveats. Streaming to the Chromecast Audio required a premium subscription and the first-gen Chromecast wasn't supported. That second issue won't be a problem anymore. Reboot your first-gen Chromecast, and Spotify should work just fine.
The lack of first-gen support wasn't some dastardly plan to get you to buy a new $35 Chromecast. The old streaming stick simply needed a firmware update to work properly with Spotify. Google will still sell you a new Chromecast, but there's little reason to buy one right now.
Your device might have grabbed the Spotify update already, but if not a quick reboot will get it done. The Chromecast can be rebooted from the Chromecast app or you can just unplug the power.
- Kristoffer and George