You can check to see if the pulley is loose by drawing a line across the pulley, nut, and shaft with a Sharpie. Run the engine a little and see if everything still lines up. That would be an easy fix.
If it were my car, I would start with charging the battery. Then start the car and check the voltage at the battery. Turn on your lights heater blower, radio, etc and check it again. A fully charged battery should be at 12.5-12.8V. With the car running you should see 13.5-14.5V.
Some 'rebuilt' alternators are checked, cleaned, and placed in a new box. Unless the bearings are bad, it is usually the brushes that were out. They are very easy to replace on a Volvo.
Greg
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