It looks like the game uses the battery for saving high scores, audits, and credits in machine. If you don't need it to save any of that stuff, you can continue simply using the game as-is.

If you want to replace the battery: what i use in my game is a 1.5F/5.5v memory capacitor, bought from Great Plains Electronics here: https://www.greatplainselectronics.com/proddetail.asp?prod=CERS-1.5F-5V

You'll need to solder a wire to each of the 2 terminals on the capacitor and solder them onto the pads where the battery was soldered. Note the capacitor is electrolytic, and has a + and - side. You want to be sure to connect the + terminal to the + battery terminal on the board, then the - terminal of the cap to the - battery terminal on the board. Then you can use hot glue or something to fasten it to the empty spot on the board where the battery was. Thought this isn't from my sharpshooter, I'll include this for a visual reference: IMG_20200203_160331.jpg

One caveat: memory caps need to charge up for a while (the game needs to be on for a couple hours) before they start keeping memory, and once working will continue to work as long as the game is turned on periodically (every couple months or so)

As others noted, NVRAM is another good option, but I'm not sure if it's available for Game Plan games (they might be though)...

Good luck! Very cool game, i hope to play one someday