I assume it's quite heavily automated. Certainly, organisations like IGS [1] operate reference networks where most of the base stations run unattended - you buy something like a Leica GR25, connect up the antenna, power and ethernet, and let it get on with things. I don't know whether the upload-to-satellite process is 100% automated - I assume the 50th Space Wing [2] finds something to occupy their time! And naturally base stations need some maintenance [6]
Even if the entire control segment is lost, the designers of GPS are way ahead of you - although I suspect rather than thinking of zombies they were thinking of nuclear war with the USSR.
You can predict orbits in advance - but the results become less accurate the further out you try to predict. According to [3] we can predict 7 days out and get a range error of 10m, and 28 days out with a range error of 100m.
Some of these longer-term predictions are uploaded to the satellites in advance; I've heard 14 days [4] and 60 days [5] quoted.
TLDR: Designed to be accurate enough that your submarine can pop up 14 days later and nuke Moscow. After a few months, all bets are off.
Wow, I didn't know the whole system is so fragile. I wonder if it would be technically possible to have the satellites control themselves by measuring passive fixed points on the earth surface, although I suspect the current system is optimized for cost (weight and ease of upgrade).
How automated are the control segment operations? After the zombie apocalypse, if the power grid is still up, will I still be able to use my GPS?