Monaco has another Jardin Exotique, much bigger than the one at Eze. I like them both for different reasons.
Monaco also has one of the top - if not the top - aquarium in the world, at one time run by Jean Cousteau and with some of his equipment. The rooftop views from there out into the Med and back into the Rocher are fab.
Monaco is like a 3-D jigsaw puzzle. It isn't only what you see on the surface, Monaco is linked up and down the hills and cliffs by a whole hidden network of escalators, tunnels, moving people movers, lifts (elevators) and other simplifying devices... It might take you half a day to figure out and explore all the ways out of the Monaco train station.
There's also a changing of the guard, a nice rose garden (near the helipad), an equally nice Japanese Garden (on the roof of the car park for the Grimaldi Arena), a world class football (soccer) team, and so much more packed into quite a small country.
Get over to Cap Ferrat and have a walk around the shore, and then up to ridge of the Cap to the Rothschilds place with the house and the gardens.
Villefranche sur Mer is pretty good too.
Further afield, or how much time do you have?