I have a hole in the oil cooler, can anyone tell me if I have to remove the entire radiator system in order to repair hole.

The cooler can be removed with the radiator in place. The top tank is loosened enough to allow the side support of the radiator to be removed so the cooler can come out the side.

You can also just block off the cooler lines at the oil filter base or better, loop the lines back together with soldered copper tube fittings etc. The tractor won't miss the oil cooler, especially if you take the opportunity to remove the top tank completely and rod out the core. If you want to do this, I would recommend doing a radiator flush first with muriatic acid at about 1 part in 10 parts water to soften up the scale in the flues. Rinse completely afterward, but I expect you will still find plenty in the top tank when you take it off.

Just be careful of the bolts threaded into cast iron. Loosen carefully and perhaps go back in a turn or two if one seems to get locked up.