1. How do I go about inspecting the alternator to find out if this is indeed the culprit of my problems? Detailed very well above. Napa can do it if you like, with a load test.
2. I don't have a multimeter. Last time I was at Sears, the prices ranged from $10 to $200. What functions do I need on a regular basis? I prefer to buy good tools that will last a long time and be accurate, so if the $90 meter will perform more accurately and reliably than the $10 tool, I'd prefer to buy the good one. However, I don't want to spend $200! Like Jeff says, you only need the most basic functions.
3. If the alternator is the problem, can I rebuild it rather than replace it? Yes, it's simple on a minitruck and I would expect it to be similar in an 80. I wrote this, which also covers replacing the brushes, a common issue and very inexpensive to replace. Not sure how much is directly comparable to FJ-land.