You didn't specify in your post whether the PLC was the 5/30 or the 5/80. I think we are assuming it is the 5/80 since you said you can see it in RSLinx. If it is instead the PLC-5/30 you are trying to talk to, please let us know.
You mentioned there is no fault light on the CPU. I assume you are saying there are no red lights on the CPU. If the keyswitch is in Run, you should see a solid green PROC light. If it were faulted, it would be flashing or solid red. RSLinx seems to think it is faulted.
But as Ron pointed out, not seeing a name for that CPU would usually indicate a fault where the controller has dumped the program. That would show a solid red PROC light (non-recoverable fault) on the CPU. If that is the case then you cannot go online with the controller since there is no program loaded. It cannot recover from this, The program must be reloaded.
If the key is in the hard run position, the fault cannot be cleared and you can't reload the program. Move it to the REM position and the software will be allowed to clear the fault.
If the light is flashing red, then the program is still there but there is a recoverable fault. With those, the existing program is intact so you can recover from this. But, by all indications, this is not the case.
OG