Try this: Go into your settings and allow apps from external sources. Then download the Spotify apk from herehere to your tablet, and run it. No guarantees it will work, but if it doesn't, it's not going to break anything.
Edit: It won't create the link for some reason, so here's the URL:
www.spotify.com/wp-content/uploads/android/0.4.04/SpotifyAndroid.apk
Another option is to find the apk on your phone, copy it to your tablet, and run that. It looks like it's just a single apk for both tablet and phone.