No need for a PM, I sniffed out the thread (ok, so how many times do you see VW in an FJR forum?)
Steve, being that it's an 04', my first guess would be the battery. Most OEM batt's are good for 5 years, but I've seen them croak in 2. Heat does very interesting things to electronics, creating extra resistance is one of them. The Batt can show a nice healthy 12v, but most volt meters just show the surface charge. Take the car to a local Autozone/advances auto and have them run a diag on the batt/alt/starter system. They will do this for free.
Being that you get 13v+ when the car is running, does lead me to believe that the alt is good. The problem can be in the batt or the starter since the starter selenoid can be effected by heat as well. However, I'm leaning on the batt in this case. I had a similair issues 2 years ago with my 01' Jetta. Still had the OEM batt in it and cold started fine. Wife made a short run into town and couldn't get the car to start when she tried to come home (less then 10min at the store). Even though my voltmeter showed the exact same things as you saw, I replaced the batt just to be safe (was past it's freshness date anyway). Problem solved.
If it's the starter (which I've had to replace on my daughters 98' beetle) then it's really not too bad if it's a 5 speed. I was able to change it out without having to even jack up the car. However, I did find things went much smoother after removing the battery & airbox. However, that in itself is a royal PITA. Would like to shoot the engineer that designed the engine bay in the beetle. If the wifes car is an automagick, I haven't tackled one of those just yet, but I'm sure I can find a nice how-to for you.
Feel free to PM me any questions just incase I loose track of this thread. If it's found that the issue isn't in the batt, starter, alt, please let me know. I'll see if I can locate some fellow dubbers in your area that might have a Vag-com tool. It's basicly the exact same software that the dealer will use to diag the car. I have one, but I'm a little far out to make a quick run down.