Question

My uncle Robert told us about the time when he backpacked across the island of Luzon.

Answer 1

My uncle Robert told us about the time when he backpacked across the island of Luzon.

That part of the sentence fuctions as and adjective since it is modifying a noun.

Adjectives are those words which describe or modify other words(nouns) , making your writing and speaking much more specific, and much more interesting. They are usually positioned before the noun or pronoun that they modify.

Answer 2

I'm not sure what exactly the underlined words are, but I'm assuming it's of Luzon. 
In any case, the correct answer is adjective.