Yes, whatever will cause the most damage needs to be fixed first. Change the thermostat and while you're at it flush out the old coolant completely
and use Honda Coolant. The most important that needs to checked when you first get an Integra is the timing belt. How many miles does the motor
have and when was the last time your friend changed it?

If it needs to be changed, you can do the thermostat & water pump at the same time. I recommend Valvoline high-mileage oil, its a great non-synthetic oil.