I have no cold water at all in the bathroom and extremely low pressure in the kitchen. The house was built in 1939 and the problem is in the second floor apartment.

Is hot water OK? Is the piping galvanized or copper? Do you have good pressure on the first floor? I wouldn't be surprised to find a 70 year old pipe at least partially plugged. Is there a shut off valve to the second floor that may only be partly open? There could also be sediment clogging the sink faucets or shut off valve; turn off the water under the sink, remove the riser pipe at the faucet and slowly turn on the water-you'll need a bucket-. If no water comes out your problem is in or before the shut off valve. If a good flow of water comes out, you may have a plugged CW stem or valve depending on how old the faucet is Next, reassemble the piping and remove the CW faucet stem and then open the shut off valve slowly and flush the faucet.