Your household voltage is nominally 120V/240V. The US hasn't specified 110V for a very long time.
Yes, that transformer will work (you could easily use one of the smaller versions), but remember that it will step up only the voltage, but will not alter the line frequency. (US is 60Hz, many countries are 50Hz.) Most devices don't care, but some do and will malfunction.
The only part of that desoldering gun that might actually care is the pump, potentially. I'm no pump expert, but just remember that those pumps work by reciprocating the membranes at line frequency. My guess is that it'll probably be fine, but may have a slight change in suction, since it'll be moving 20% faster than specified, but the inertia and drag of the air itself might make it more or less efficient per stroke. (This is pure educated guessing, nothing more.)
I still don't recommend deliberately getting devices for another power standard. Instead, as much as possible, try to get universal devices, and if that's not possible, get something local. Transformers waste energy and take up space, and really only make sense for temporary use or for unusual devices you simply cannot get for your power standard.