Is that dish it is sitting on filed with water? If so, remove it immediately, because you are killing it. Citrus must never, ever, ever sit in trays of water. What this tree needs now is some tlc. Give it plenty of good light, high humidity and warmth at the roots - the roots should be about 70F if not more (this is the temperature in the pot, not the air temp). Keep it away from that radiator. You should let it dry out a few inches down in the soil before watering again, and when you do, water deeply so that a good portion of the water drains out of the bottom. Avoid fertilizers until it is showing signs of good health again.
So, be vigilant in watering, learn about watering citrus, put it in good light, keep the roots warm and give it heaps of humidity
As for the potting mix...I don't know whether I would repot it now. It might be stressed, but if the soil is really that appalling, repotting now might be worth the risk. Let an expert chime in first though before you do.
I admire you for buying the tree in its run down state. I for the most park buy healthy plants, but I also enjoy buying the more sickly ones, just for the joy of nursing it back.
One other thing to do is check the tree THOROUGHLY for pests and diseases. Every stem and leaf needs to be inspected.