How to attach newel post to stringer below the tread?
I'd use four 5/16" x 3-1/2" lag screws through the stringer into the post, properly piloted, and construction adhesive at all contact points. Spread the screws as far as possible, staying at least 3/4" from any edges, for the best stability.

The bigger concern, though, is the attachment of the stringer to the landing. If that's not robust, your post won't be, either. If possible, add construction adhesive to that joint at all contact points, even inside and around any builder hardware (such as a hanger). Even a heavy bead wrapping the joint and tooled out slightly will add significant strength. You could also add a block to one or both sides of the stringer, against the hanging surface, and embed that in adhesive. Anything along those line will really help give you a solid post.