if it run for a long time with a leaking oil cooler , it could have done lots of damage. ( if overheated without oil in the engine ).
take out the cap where you put the oil and have a look under the cap itself. If you find old oil remains or white soapy stuff this means that there is something wrong with the engine. Either you have a broken or non working piston ring, or for the white soapy stuff there is water in the oil.
In both cases this is bad news.
In the case where there is no leak from the oil cooler and the oil inside the engine is ok, check the radiator water, start the car and let it warm up.
Turn on the heater inside the car and wait for the radiator cooling fans to start.
If the fans start at about 83 - 87 degrees and you get hot air from the heater inside the car, this means that you probably have a clean cooling system. If everything is ok, means the engine should be just fine.
Hope this was helpful , good luck