Hello,
I had the same problems a while back with my 1990 Accord EX. I replaced the battery and that wasn't the problem. I bought the repair manual "Chilton's Honda: Accord/Prelude 1984-95 Repair Manual" (See if this link works:
http://www.amazon.com/exec/obidos/tg...glance&s=books ) and used it to see the best way to replace the alternator. You can buy it at your local parts store for about $15. Of course, some of the photos aren't great, but if you study where it is on the engine and what it is connected to, you should be able to figure it out easily. I used to get overwhelmed at the thought of doing such a thing, but I just had to jump in there and try it.
You will have to take off the powersteering pump, but that isn't very hard. Be careful to drain the power steering fluid so it doesn't spill.
Just keep track of what parts went where and you should be fine. I felt mighty proud of myself after replacing it and it fixed my problems.
If you need more direction, let me know. I am happy to help. I am learning too and it is encouraging to find others that have done the same repairs.