The one shown is an autotransformer, there are 2 varieties I have met. the first is a 220v winding with a centre tap, and the other is a 55VAC, 110VAC,55VAc winding variant. The first will either have the one side at neutral or line voltage and the other wire 110V offset from that, the second will always have 55VAC on one leg and the other at 165VAC, giving 110VVAC between them.
How do you know the one shown is an auto transformer? Do you have x-ray spectacles or something?
It's more likely it's an isolation transformer, especially given its size which is quite large for an autotransformer, which would be half the size for the same power rating and the fact the connections to the primary and secondary are on opposite ends of the case.
I'm confused now
With a short answer can i do what i described or i will burn my house down?
What you've drawn is fine.
In addition to that, if it's an autotransformer, it's important you've got polarity of the the phase and neutral correct on the primary, otherwise the secondary will float at 220V.
An auto transformer only has one winding. The neutral should be connected to the common connection which goes straight through to the load. If the phase is connected to the common instead, the voltages on the secondary will be 220V (which should be 0V) and 330V (instead of 110V). This wouldn't blow up the soldering iron but it would put extra strain on the insulation and could pose a shock risk if something goes wrong inside the iron.

On the other hand, if it's an isolation transformer it will have two separate windings, so polarity of the connections is not critical.

It should be obvious whether you've got an isolation transformer or an autotransformer when the cover is removed. An autotransformer will only have three connections to the windings. An isolation transformer will have four connections: two for the primary and two for the secondary. This is assuming of course there aren't any other taps or windings for different voltages.