New GFCI outlet on ungrounded circuit won't trip when tested
The test button on the GFCI should work it doesn’t use the ground. The tester uses the ground with a resistor to create a ground fault since the ground is not there the test function will not work with the plug in tester. As long as the test function on the receptacle works the receptacle is protected as required by code.  

The electronics compare both the hot and neutral, a imbalance of more than ~5mA will cause a trip. This is a legal method of GFCI protection with a 2 wire circuit.  

IF you want to run a ground wire back to the panel that feeds this circuit OR if there is a grounded circuit in the area that is fed from the same panel recent code changes allow a new separate ground wire to be installed. 
However if the test button on the GFCI works the receptacle is protected. 

Just a note, in my jurisdiction a rental requires a licensed electrician to do anything more than replace a device (light, receptacle or switch) for example. I have heard of some that require a license for even those but have not worked in a state that has that strict requirements. I would verify your local regs so you don’t get in trouble.