"Please climb up that mountain."
Do they distinguish in Japanese between 'climb the mountain' and 'climb up the mountain'?
When using "kudasai" after a verb, the verb is in its "te"-form. So のぼって is the "te"-form of the verb のぼる, and you add ください to make it a request.
Is there an issue with using the particle を instead of に? One of the first example sentences here on Jisho.org uses を when referring to "climb a hill": https://jisho.org/search/%E3%82%92%E7%99%BB%E3%81%A3%E3%81%A6%20%23sentences