Learn Russian: По прибытии в гостиницу вам нужно предъявить паспорт. - Upon arrival at the hotel, you need to present your passport.

This formal structure uses по + Prepositional case to mean 'upon' or 'immediately after'. The dictionary form of the noun is прибытие (arrival). Neuter nouns ending in -ие take -ии in the Prepositional case, which is why it becomes по прибытии rather than taking the standard -е ending.

Why is it в гостиницу (Accusative) instead of в гостинице (Prepositional) if they are arriving 'at' the hotel?

Even though English uses the static preposition 'at', the Russian noun прибытие implies motion towards a destination. Therefore, it dictates the Accusative case for the destination (в гостиницу), exactly like the verb it comes from: прибыть в гостиницу (to arrive at/into the hotel).

Why is 'you' translated as вам (Dative) instead of вы (Nominative)?

The word нужно translates to 'it is necessary' and forms an impersonal construction. The person who needs to do the action is the logical subject and must be in the Dative case, so 'you need' is literally expressed as 'to you it is necessary' (вам нужно).

Why do we use the perfective verb предъявить here rather than the imperfective предъявлять?