Welcome to the forum.
I don't have any dimensions for you, but I have several ideas as to how you could approach the scoring.
1. Build a control circuit that takes inputs from switches in each score tube to add the apropriate number of points to a readout. It would take a reasonable amount of electronics understanding, but the parts would likely only be about $30.
2. If you have a spare computer sitting around, or look for a free old one on Craigslist or similar places, you could hack a keyboard or joystick/game pad to send inputs to a simple program that added points to a running total and displayed it on a monitor or LED readout.
3. Same old computer, use an encoder from
http://www.GroovyGameGear.com, or
http://www.ultimarc.com to do the input and a monitor or led readout to display the score. Still requires some custom software.
4. Hunt down an old dead skeeball game and steal the electronics out of it.
All those options would need refinement, and I've not done any of them myself, but none would be too difficult. The cheapest options depend on what you have laying around, and how resourceful you are.
It sounds like a fun project and has sparked some ideas for myself that I'd forgotten I had. I don't have room for a skeeball machine (but it would be cool) but I once wanted to make a scoring system for a game (sporting) that my friends had invented and think it may be some fun circuitry to learn how to do. Thanks for the reminder.
Good luck.