Quickleaf
Legend
They have very much the chance to not find the gate, and if they do find it they are probably in bigger trouble since inside the gate there is a Fire Giant and his Hell hounds guarding a entrance to the Underdark, if they manage to convince the Fire Giant that they are trustworthy they gain a strong ally but if not there will be one hell of a fight, besides that my players are super curious and will probaly try to go to the Underdark even if they are a bit underlevel to go there in my eyes
Gotcha. So it sounds like the stakes during the mountain climb are: Will they find the golden gate?
In that case, think of the mountain climb session as being all about the search. They know it's somewhere on the mountain, but not exactly where. So they're hunting different areas of this mountain, searching for clues which point to the golden gate.
You can see a skill challenge I put together for a similar concept (searching a jungle wasteland for a bone palace) over here: [5e] Help with a travel skill challenge? Most of the work went into designing a sort of logic puzzle which the clues helped guide the PCs toward solving; so it was a process of elimination – "Ok, it's not hexes A, B, C, D, G, or H based on this clue, and this other clue tells us it's not hexes K, M, or N, so we know..."