Changing radiator hose. Do I need to worry about after? Leave the radiator cap off to prime?

First time replacing the radiator hose as I’m getting antifreeze spray from the area and thought I’d go cheap initially. I’m also replacing the hose to the overflow box as it’s wet and I have antifreeze spray around the cap and on the actual box. Do I leave the radiator cap off after install to remove the air? 2000 3.4L. The truck is not overheating but the overflow is draining quick. The radiator fluid is full.

The radiator is about 13 years old and I haven’t seen any leaks so my thought is if I replace the hose, clean up the spray, if it’s still blowing antifreeze, then I’ll have the radiator tested.