For the replacement battery you'll need a CR2032 battery with pre-soldered tabs. This is the direct replacement for your old N64 cartridge battery. This is the battery I used and it came with a 3.8mm security screwdriver, which you'll need later.
Unless you have a pulse spot welder laying around, don't solder anything directly to the battery.
As always, Nintendo loves to try and keep us out of their hardware. You'll need a 3.8mm security screwdriver to open the cartridge. Use it to remove the two screws on the back and open the case.
Remember the first video game you played? (Note...the answer will age you!) For many, video games stay in our memory as markers for certain periods of our lives—as long as your old enough.