I measure the thickness of the wall with a simple bent wire "caliper".
The gap remaining in the wire lets me know how thick the wall is. That is, the original gap minus the remaining gap equals the thickness. I am not worried about actual measurement so much as consistency. As long as I am less than an eighth of an inch I know I am good for the weight.
The cap for the ornament has a 3/8" tenon. I use a 3/8" bit in a Jacob's chuck in the tail stock to drill the hole. To do so I reduce the lathe speed to 950, drill the hole remembering there is only a wall thickness of 1/8" or less, and then return the speed to 2500. I think it is faster in this case to use a belt drive rather than a variable speed lathe, but opinions vary. The lathe you own is the best one to use.
Part off the ball remembering that there is a 3/8" gap in the center. If yu forget, the ball can fly off, bounce hard and ruin that finish.There is usually a bit of waste left on top of the ball that will need to be removed later.
It is a good thing to check for fit before gluing. A bit of sanding may be needed to get everything in place without forcing.