I remember learning about hyperbolic and spherical geometry in middle school, and that was cool. Not really because of the axiomatic aspect, but more of the "how many lines can you draw through two points? Up to you!" sort of questioning of mathematical assumptions and the funny diagrams. The Fano projective plane model was interesting, the Poincaré disk model was interesting. I remember some animations and some interactive software you could play with. But yeah, after about a week I got bored of it. I would say one 2-hour session devoted to different geometries and constructions would probably be about right.

The Euclid's Elements approach of axiomatic geometry is interesting, and suitable for maybe a high school course. Before students learn algebra they don't really have an appreciation of deriving equations or proofs from a small starting point. And coordinate geometry is much more practical (some things are simply unconstructible with ruler and compass).