While you can download podcasts to your Apple watch and play them while the iPhone is away, Spotify however does not support offline playback on the Apple watch.
See here: Listen to music, podcasts, and audiobooks on your Apple Watch - Apple Support
So yes, you will need a cellular watch to stream music from the internet.
Unless your watch is the cellular model, (there are 2 different models), you cannot add cellular to the Apple watch.
You will need to pay extra to have a GPS+Cellular Apple watch on your iPhone service plan yes. Around $10, but depends on the carrier.