Radiator question

Long story but here is the condensed version. The car had the original engine when found in a barn. The owner got the engine to run but he said it kept overheating. He switched the engine out for a 1969 engine. I buy the car and want it to be as original as possible so when I purchased the car I had the original 1965 engine rebuilt. Once installed I noticed the engine would run hot by the gauge. It would pin all the way to the right. Then it will back off but never in normal range. I have ordered a auxiliary temp gauge and temp gun but if I need to clean water jackets while the engine is in the car what is the best way to do such. Has anyone got any ideas?

If the engine was rebuilt the water jackets are not the problem.