30 W 220 V solder - How to run on 110V?
That circuit is called a "transformer", and you wouldn't build it, you'd buy it.

Yes, it's absolutely not feasible financially to build or buy one just to run a $10 soldering iron, but it's really just two runs of cable wound around the same iron core. One side has twice as many windings as the other – that's the side with twice as much voltage.

Of course, modern transformers that you can buy have very cleverly designed iron cores to minimize losses, they're very diligently wound and isolated, encased and contacted, but the basic principle is that both sides are electromagnets: the primary side converts an alternating current to a time-changing magnetic field, and the other converts a time-changing magnetic field to an alternating current.

The wikipedia article on transformers will be of interest to you.