If there are no typos, наверну is a verb (1st person future tense) which can have different meanings (technical - to screw on or slang - to eat, to dig in). Другой could mean a different / another one / one more.
However, the question title makes me think there is a typo, and the phrase is:
Наверно is вводное слово gramatically, meaning probably in English.
So the full sentence is:
У него два туалета - один внизу, другой наверху.
Другой is an adjective, nominative case. The noun is omitted to avoid repetition, but you could also say:
У него два туалета - один туалет находится внизу, другой туалет наверху.
Наверху is an adverb, meaning upstairs.