The GBA stashes the palette of a given game in memory (detailed on the links above if you want) and uses that to choose colours.
The most basic thing to do is dump this palette (savestates will have it, some tile editors will even fish it out from a savestate, some emulators will allow you to spit it out as another format and you can always just dump memory) and search the ROM for this palette. This works a surprising amount of the time as palettes are usually very small so tend not to be compressed.
The trouble comes with palette animation and dynamic palettes. Palette animation is much as it sounds and the palette is changed to effect an animation.
My usual two examples are Mr Driller 2 and various Summon Night games. For the former then the rainbow block is generated by rapidly cycling a palette colour. For summon night then there is a save point that is a glowing sword, bit more elaborate as it is a fading gradient effect but same deal really.
Summon Night 2 also provides a nice example of the dynamic palettes concept. Here the options to change the colours for GBA, SP and GB player (GBM and 101 were some time out yet, indeed I don't know if any games have options for either of those) and it does this by changing the palette entirely (believe all in rather than 3 palettes for every item).
For dynamic palette you can often search for the fragment before or after the changing bit and get somewhere. The more dynamic stuff might still get something or some might have some fun with relative search (if everything is +10 across the board...) but will probably be a bit harder.
"with the original color palette"
Careful with a search term like that. For GBA purposes all those early SNES ports often got a brightening (read washout) to deal with the stock original GBA screen, which was a dubious idea even back on the original GBA but enough of that, but the SNES and GBA palettes are the same format. This is what you will tend to find if you go searching for that. As Golden Sun was not on the SNES there is no original palette to go back to, though from context you obviously mean colour match the game on a 001 screen or something similar.
A basic worked example of a basic palette hack can be found https://web.archive.org/web/20080408022631/http://etk.scener.org/?op=showtutorial&st=3
Some more advanced stuff can be found in http://www.romhacking.net/forum/index.php?topic=14708https://www.romhacking.net/documents/361/
is for an older emulator (would suggest no$gba these days as the debug version went free a while back) and for graphics rather than palettes but the terms should carry across and one is a dedicated area of memory that the game writes to so it has something for a task and the other is graphics RAM so no great shake there.
You can also come the other way and do graphics hacking -- assuming it is just a single use sprite (no palette swaps or it does not matter for your purposes) and there is another colour in the palette that works for you can change the graphics themselves to use the other colour already there.
I should also say Golden Sun saw a considerable amount of hacking work done to it -- depending upon how you measure it then it might even the most popular to hack GBA game. I don't know if Atrius' fantasic little tool ever gained palette editing options but it would not surprise me in the slightest. Failing that there is probably a nice guide somewhere on the matter.