We can't advise you enough to go up this hill.
It tops at 90m and offers a magnificient view on the city in all directions. Not as easy to access as the Promenade du Paillon, it will reward generously those who will have taken the trouble to go up there.
A maze of paths and stairs between trees and scents. A landscape changing after each step, while you go up. The quietness of a park with no cars. A children's garden. The Castle hill will offer you all this at the same time.
That's indeed here that the Nice castle was located in the Middle Age. There is very little left of it. The ruins are at ground level and that's not the reason why you should come here.
At the top, a terrace facing West gives you a wonderful view on the Promenade des Anglais and the mountains in the background. Courmette peak (1200m) , Cime du Cheiron (1800)...
Eastward, behind the children garden, an unobstructed view on the harbour, the Mount Boron, the Mount Alban and the Cape of Nice. Much further, in the gap, the Mercantour summits, snowcovered in winter (2500 to 3000m).
If you don't mind walking a little, the best solution is to go up with your feet. Several possibilities from the Old town through stairs and narrow streets. Your children will love it, especially if you can't follow them... You can also access the hill from the harbour side following the road. Not the best solution because it is a little narrow, with no sidewalks and you have to move to the sides when a car passes. There are also several stairs and paths on this side meaning that you can avoid the road if you don't mind a steeper climb. Last possibility : the stairs starting from the sea shore, Quai des Etats Unis.
There is also a lift going more than halfway, starting rue des Ponchettes (on the Sea side).
By car the only way up is the Montfort alley, on the harbour side.
In any case you'll end up walking. Cars are not allowed beyond the cimeteries. There are not many parking spaces and moreover you'll have to pay to park your car.
On the parking, a large memorial to the "Justes", those people from Nice who saved Jews during WWII.
To go to the top you'll probably pass along the fall (see above) which can be seen from everywhere on the Promenade des Anglais.
Your children will love the garden which is meant for them, with plenty of games. A field beside is used to play balls or to run.