Which Dyson models does this battery fit?

It is a model-specific part, so check your vacuum's model number against the 969352-03 listing before buying. It is not a universal Dyson battery.

Will it bring back the runtime my vacuum had when new?

A fresh pack restores close to original runtime if the rest of the vacuum is healthy. If the motor, filter or brush bar is worn, runtime and suction may still fall short.

Is it better than a cheaper third-party battery?

The manufacturer pack matches the original fit and charging behavior. Third-party packs can work but vary in cell quality, so lifespan and runtime are less predictable.

Is fitting it a job for a professional?

On the models it is made for, the pack is designed to be swapped by the owner. Follow Dyson's instructions for your model and dispose of the old pack properly.