Learn Arabic (Modern Standard): حسب الخريطة، الفندق قريب من النهر. - According to the map, the hotel is near the river.

In Arabic, the adjective قريب (near or close) needs the preposition من (from) to connect to the noun that follows it. You literally say the hotel is "close from the river" (قريب من النهر). Leaving out من is a very common mistake for English speakers!

Is حسب a verb here? I've seen it mean "to calculate" before.

It shares the same root, but here حسب is acting as a fixed particle or preposition meaning "according to" or "depending on." You will see it used very often at the start of sentences to introduce a source of information, just like in حسب الخريطة (according to the map).

If I am reading this sentence out loud with full vowel endings, what grammatical case does الخريطة take?

It takes the genitive case (majrur) and would end with a "kasra" (an "i" sound: الخريطةِ). This is because the word حسب behaves like the first part of an Idafa (a possession or relationship structure), which forces the noun immediately following it into the genitive case.

Could I use the word بجانب instead of قريب من?

You could, but it changes the meaning slightly. بجانب means "beside" or "next to," which implies the hotel is immediately adjacent to the river. قريب من simply means "near," so the hotel could be in the general vicinity but maybe a couple of streets away.