Yes, the word year-round is an adverb (it can also be used as an adjective, by the way) which modifies the verb enjoys. It tells you how the city of Kathmandu enjoys its warm climate—it does that all year long. Meaning, the place has warm weather conditions throughout the entire year which is what, according to the person who wrote this passage, makes living there such a pleasant experience.
You should remember that in English adverbs can only modify verbs (she talks loudly), adjectives (he is very tall.) and other adverbs (the show ended too quickly). They never modify nouns, compound nouns or noun phases. So, locutions such as beautifully girl or largely house don't really make a whole lot of sense. In your case, a warm climate is a compound noun. Therefore, the adverb year-round can't possibly modify it. The verb enjoy is the only thing there that it can go with.