How long do GBA cartridge batteries last?

The CR1616 battery in that cartridge is said to last about 10-20 years from manufacture. This means that, if you have the original battery, there's about a 50% chance it is dead right now, or will be very soon. This information was obtained from figuring out the type of battery found in that cartridge and this information, which includes that battery under GB/GBC specs (but presumably, GBA should be similar in characteristics). You should probably consider replacing the battery soon if you intend to save. Note that removing the battery will immediately erase/lose your saves without specialized equipment to either keep a charge or a way to keep the data from degrading (e.g. using extreme cold to slow the decay rate). You can also back up your cartridge data with a reader like Submodule GB01 or others.

In addition to phyrfox's great answer, it should also be noted that games which have a real-time clock tend to drain their battery at a faster rate (due to the energy consumed to run the clock) than games which just use battery-backed RAM for saving.

However, some games such as Pokémon Emerald have a battery to run their real-time clock but save into flash memory. When the battery dies on these games, your save is not erased but the clock functionality won’t keep time with the GBA off until the battery is replaced.