Minimum ampacity for circuits in a home from 1964
It is possible that the circuit that you are wishing that you could connect an AC unit to may very well be equipped with a 15A breaker and 15A wiring. However it is possible that this circuit services three users and so the landlord is telling you that it is limited to 1/3 of that capacity to your outlets on that shared circuit.

It may even be possible that the main wiring panel does not even have circuit breakers and instead equipped with fuses. It is old enough where that is a possibility. And 5A fuses were not unheard of.