What reasons make you suspect it's the RAM?
I think it's more likely a CMOS battery issue, because you mentioned the computer is old, it was shipped without power, and you had to reset the system clock.
A blown fuse also seems unlikely given that you succeeded in charging the battery. Actually, I can't recall seeing a fuse inside the cases of a VAIO before. Opening them up is tricky, as there tend to be hidden lugs that break off is you're not sure what you're doing, but you'll find it easier if you have access to a service manual.
If the laptop is very old I would seriously question whether it's worth sending back for repair. Laptop screens tend not to last forever and are prohibitively expensive to replace. You might get the current issue fixed only to find something else breaks a few months down the road. How old is your VAIO?