I do have a working Asteroids upright. I seem to get the impression that I can maybe use it to test various parts on BZ?
Yes, a working Asteroids can be very helpful. Its monitor can be interchanged with BZ to help isolate problems. (It may or may not require a simple Molex plug adapter, depending on the exact monitor models in question).
Working with the monitor scares the shit out of me because I keep reading that it can kill you if you are not careful.
Mostly urban myth. Don't work on it with any AC power connected, and it can't kill you (only thoroughly piss you off).
How much money do you think I will need to invest in tools to trouble shoot all the various parts?
That's a loaded question. Depends on how clever you are with cheap tools, how patient you are, and how certain you wish to be before replacing a given component. You could easily spend a couple thou, buying a scope, logic analyzer, high-end DMM, setting up a test bench, etc. But I think, since you've already got a working vector monitor, you could likely get by with just a good DMM, a logic probe, and the diagnostic tool between your ears.
Can a average dude pick this up and succeed or do I need to goto electronics school? I am fairly smart (Computer Engineer) so I am hoping I can work through this with a little guidance
If you took a course in basic electronics, and understand AC & DC, ohm's law, and basically how diodes & transistors function, then you're pretty much set with regard to background education.
I don't have a multimeter, but I hear they are cheap. I have a solder gun and can solder, but I consider soldering a bit of a art in itself, I could easily see myself seriousy f*kn up the original board "trying" to resolder various parts.
A multimeter is the most fundamental tool. There are lenghty threads on the topic, but I chose to spend a bit more here for a quality DMM.
If you have an actual soldering "gun", vice a soldering iron, it's not appropriate for PCB work.
Here's another reading recommendation on basic tools and techniques:
http://www.pinrepair.com/begin/index.htm
But we're jumping ahead, talking about soldering. All of the initial stuff is done with your eyes, hands, and a DMM. We have to identify a bad (or likely bad) component before breaking out a soldering iron.