All fixed appliance are required to have a dedicated circuit but given the age of your house its not surprising that it isnt. At a minimum your LL should have had the other outlet removed and blanked off.
Your LL should get an EC out to look at it but if he wont I would recommend you do get one your self, depending on where you live a 1-2 hour service call will run you about $150-$300.
Your AC really should be on its own circuit but not much you can do, if I had to use an extension cord I would use one thats atleast 12 or 10. I also recommend you buy an outlet tester and verifiy that all the outlets are wired correctly or If you have an EC come out they can do that fairly quickly