92 Chey Lumina Won't keep a charge

So my car was making a ticking nosie to start up yesterday and today I had to get it boosted to get it going, but as soon as its turned off it won't turn back on. The alternator looks faily new. I just bought the car a few months ago. Could it be a battery issue or is there things I can do to test to see what it is before I take it to get it fixed.

Also is there a likelyhood that a car with this problem would keep a charge while driving for an hour so I can get to my normal mechanic or should I not chance it and get it fixed where I am?