How to fix a newel post cap

Buy an 8mm dowel in a hardware store.
Take cap off and clean both surfaces thoroughly including all paint from the edges.
Drill a hole for a dowel in the centre of both the cap and the post itself. Ensure hole is centred and exactly on vertical axis for both pieces
Use wood glue on both dowel and flat surfaces.

See if you can fashion a weight of some sort to keep downward pressure on the cap until the glue dries but make sure the weight isn't putting applying side-ward pressure also.

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Buy an 8mm dowel in a hardware store.
Take cap off and clean both surfaces thoroughly including all paint from the edges.
Drill a hole for a dowel in the centre of both the cap and the post itself. Ensure hole is centred and exactly on vertical axis for both pieces
Use wood glue on both dowel and flat surfaces.

If it's in a location where it's taking weight (people grabbing it for support, which is presumably why it broke in the first place), I'd do the above, except using a steel rod instead of a wooden dowel. Considering it's painted, rather than trying to drill both parts dead centre separately though, I'd drill down through the round top of the newel post with a long bit, and continue drilling into the square section. Once drilled, get glue into the joint, hammer metal rod down through both, filler in the top of the newel post to patch the remaining hole, sand and paint.