It is possible but it is a lot of work. You'd be better off buying a flash cart. Probably cheaper too.
What is involved though. Well first of all you'd have to desolder the mask ROM chip (the IC that contains the game) and then you'd have to solder a flashRAM chip in its place and you'd also need a custom programmer that could interface with the GBA port on the cart and also would be able to program the chip through the circuitry that is there.
You'd end up having to build a flashRAM chip onto the cart and also would have to build a programmer. I've done it before with GB/GBC, but I haven't heard of anyone making one from scratch for GBA.
Or I guess you could use an eeprom programmer and program an eeprom and solder it right onto where the mask ROM was. But I dunno why anyone would want to do either of these two things. It'd waste a lot of time.