Quoted from ForceFlow:

That said, the "electronics salon D-269" board is a shield for an Arduino Mega.
(Warning--the video on this page auto-plays): https://czh-labs.com/products/electronics-salon-prototype-screw-terminal-block-shield-board-kit-for-arduino-mega-2560-r3

So essentially, you are missing the processing unit that would (presumably) contain all the code to run the game (if it was finished). And that plugs into the shield board (the D-269 board).

A shield board is basically an adapter between an arduino and something you're connecting the arduino to.
I can't quite read a make and model on the relay boards, but if you google that, you'll probably come up with more info about those boards and how they operate.

This is really beyond the scope of electromechanical game repair, and crosses over significantly into custom/homebrew pinball.

This might help a little bit, but the site is mainly focused on more modern electronic game platforms, rather than retrofitting electronics into an EM game: https://pinballmakers.com/wiki/index.php?title=Main_Page

If you're serious about tackling this project, you'll need to do a pretty substantial amount of research, have to have a pretty good understanding of EM games, as well as a good understanding of arduino platform, and additionally, be able to figure out what the previous game owner was attempting to do, and figure out if things were actually done correctly in a way that could actually operate the game.

All I can say is I wish you the best of luck if you decide to tackle this project, but I wouldn't recommend it, especially for a beginner.