The castle per se is not that amazing, but it offers you an amazing view over Zagreb (It's a little far away), and it's worth the journey, because it's not the easiest place to get too. You can smell nature up there and it's also very quiet and calm.
I leave you some details about how to get there, and some photos. They should a little movie about the castle.
Admission fee: 15kn (+- 2€), - CLOSED ON MONDAYS -, rest of the days from 11am to 6pm.
How to get there (I'm gonna give a couple options, assuming you are in the town center and you don't have a car):
Catch the tram number 08 or 14 to MIHALJEVAC (You need to exit in the last stop which is MIHALJEVAC).
From MIHALJEVAC you have two options (I did them both):
1º You can catch the BUS number 102 to BRITANSKI TRG (You need to exit in SESTINE right next to a church, and ask where the Medvedgrad Castle is) and you can walk your way to the castle, if you have a quick pace like myself you should make it in +- 1h30m (I did this with 36º, there are better things in life i can assure you, and i love to walk).
2º In MIHALJEVAC, i just take one of those Taxis that are there, and you pay less than 50kn (+- 6.70€), i paid 45kn and i'm a tourist! I strongly recommend you this way, it's super fast, cheap, and you don't break a sweat.
This review is the subjective opinion of a TripAdvisor member and not of TripAdvisor LLC.