So I found this tutorial that replaces the battery and from the screenshot he implies the game save is unaffected. It requires a few custom tools and breaking sadder connections so it is rather involved. http://imgur.com/a/pSq8B
Liability statement: I recommend getting an action replay and saving your game state to it. Then if the save does die you can restore it.
That said I doubt you will lose your save data. Data that is stored with a battery (like the original zelda 2 nes game) Are stored with a technology called CMOS that requires a small amount of power to keep it's registers stored values alive. However game boy advance cartridges use flash (or one of it's archaic predecessors) that store data permanently by "programming" a memory chip. They do not require power to run. A power surge would cause the rest of the cartridge to fry before it would erase the data. And as the power of the battery is so very small it will not surge.
All that said if you don't want to front for an action replay your probably safe to go ahead and attempt to replace the battery without losing your save file. Worst case here your replacement battery doesn't work and you still can't user time based elements. Which are mostly listed in the other answer so I'll let that answer speak to that item.