Likely has nothing to do with city water vs fresh water tank.
When you say no water do you mean zero, faucet knob doesn’t even let out some air or kick on the pump (fresh water tank)?
About impossible to diagnose a partial issue with little description start taking things apart. Like disconnect cold from faucet turn on water to diagnose obsolete faucet as issue or upstream.
start with easy one first can get to kitchen plumbing but not shower. Care of first issue may help with diagnosing the second one
Assuming private purchase because it’s old and you said prev owner. Did you ask them?