how hot is the compressor when its running? The fan went out on one a fridge of mine once, I saw that the fan was out. I just put a small fan down there blowing on the compressor and it started to work. gave me enough time to remove the broke fan and replace it with a new one.
Most likely starting components went out on compressor. You can get a 3 n 1 start kit from a HVAC parts wholesaler like Baker Dist or Johnson Supply. They don't cost a whole lot ($20-$40) and easy to install.
If the freezer is cold but the fridge isn't, it may just be frozen up. Many work simply by cooling the freezer compartment and circulating cold air from there to the fridge compartment.