If the solenoid were powered by a DC current, the ring should only experience a changing magnetic flux only at the instant when the solenoid was turned on. The ring would then have jumped only once, before falling back to the bottom. The fact that it is able to hover tells us that it continuously experiences an upward magnetic force that balances its downward weight.
This is possible only if the solenoid is powered by an AC current. We can in fact thinking of the solenoid as the primary coil of a transfomer, and the ring as the secondary coil. The primary and secondary currents are always in-phase (or completely out-of-phase if you choose). So the current in the solenoid and the ring produce magnetic fluxes which are always opposing each other, resulting in a continuous upward magnetic force (a sine-square varying force, to be exact) on the ring, which keeps the ring in the air.