Hi, I'm a sysadmin at a computer science research lab. I'd say C is probably the least important part of my job, noone in the group is a "real" C or C++ hacker.
You should be able to build programs from packaged source and be able to tweak a Makefile and library path, but for actually building administrative programs, interpreted language like shell and perl are very important. Especially since you'll endup inheriting your predecessor's tangled ball or scripts. So I would say that understanding programing genericly is a big plus, C isn't crucial. My $0.02 -Jon