The Cat IIIa certified 737s have passive autoland capability, which means that in the event of a failure, no significant mistrim or deviation from previous flight path / attitude will occur, but the landing has to be complete manually (or missed approach, of course).
As said before, with one generator inop (assuming APU generator not supplying power to the respective generator bus) the aircraft is left with only Cat I capability due to loss of redundancy of electrical system.