Typically when the Valve Cover(s) is leaking, it is noticeable, but not in an excessive amount to soak/wet the Alternator.

Also, the Covers are on the side of the engine, and the Alternator is located in the front, part of the Belt system sitting right behind the Radiator Fan assembly.

I would go to an Autozone or Advanced, they will check that the Alternator is outputting the correct Voltage to keep the Battery fully charged.

Get this checked, cause if it's not charging the Battery correctly, you'll not only be stuck for the Alternator repair, but you might kill beyond charge, the new battery you just replaced...

Also, when you put the new battery in, make sure both the posts and wire connectors are clean, very clean, and tightened.

A loose or dirty battery terminal will manifest itself as a lights out situation that you witnessed...