For the replacement battery you'll need a CR2025 battery with pre-soldered tabs. This is a direct replacement for your old battery.
Nintendo was serious about not letting people easily disassemble their hardware! To open your cartridge, you'll need a 3.8mm security screwdriver. These are inexpensive and will let you access a wide variety of Nintendo games and systems.
Remove the single screw from the back of the cartridge and remove the circuit board from its plastic housing.
The Game Boy game battery is soldered to the circuit board via two tabs. Heat up each tab and then use your solder sucker to remove the old solder.
If you haven't used a solder sucker before, it's easy -- just press the plunger down, heat up the solder with your soldering iron, and then press the button on the solder sucker and the plunger will suck up the melted solder.
If you don't have a solder sucker, you can also use a slightly less expensive desoldering wick -- but I find these much harder to use.
Using soldering "helping hands" can help to hold the circuit board but aren't necessary.
Mine was from 2003!
Tab orientation matters here -- see the attached photo as reference. Simply solder your new battery in place using a bit of fresh solder.
Use your special screwdriver to reattach the screw.
You're now ready to conquer your favorite video game, with breaks allowed -- and another ten years of save states ahead of you.
The Nintendo Game Boy is one my favorite retro hand-helds. They came in a surprising number of variations, some of which are very hard to get ahold of.