Cheap soldering irons will overheat and literally burn your Board (components will be damaged). If you plan on really getting the skills investing in a good one would be a wise choice.
Others tools/things to Invest into would bei e.g. a third hand with magnifying glass to hold the board, Flux to beeing able to solder clean, tweezers to reseat components/wires and maybe a solder Pump.
Btw.
Don't begin with your gba you will destroy it and have solder all over the place. No offense but you need skills to solder small components or wires. Try with some cheap electronics first and when you are comfortable enough you can try some complex stuff like resolder smaller ICs and FLEX cables etc. Only then I would start with a gba.