The fact that the radio is intermittent and the alternator light is illuminating is a clear signal that the alternator is not performing properly.

So, find yourself a rebuilt alternator and install it. Then see if the existing battery is sufficient to do the job.

If it's not, go down and buy a group 49 battery from Autozone. It will do the job and it carries a good warranty that is valid at any Autozone with just your telephone number. The cost was recently $70.00. Unbeatable.