Learn Hardcore Icelandic: Vinnustaðurinn er nálægt bókabúðinni.

Because nálægt is a preposition that governs the dative. After nálægt, the following noun phrase must be in the dative, so the definite dative of bókabúð is bókabúðinni. Using bókabúðin (nominative) or bókabúðina (accusative) would be ungrammatical here.

Is nálægt a preposition or an adjective here?

Here it’s a preposition meaning near and it does not inflect: nálægt + dative. There is also an adjective nálægur meaning nearby that agrees with the subject and still takes a dative complement, e.g. Vinnustaðurinn er bókabúðinni nálægur. That adjective construction is correct but more formal; the prepositional nálægt is the neutral everyday choice.

Why do we see different endings: -urinn in vinnustaðurinn and -inni in bókabúðinni?

Icelandic marks definiteness with a suffix on the noun, and the suffix depends on gender and case:

vinnustaðurinn = masculine nominative singular definite of vinnustaður.
bókabúðinni = feminine dative singular definite of bókabúð. The double n in -inni is part of the feminine dative definite ending.

What would the sentence look like with indefinite nouns?

Subject indefinite: Vinnustaður er nálægt bókabúðinni = a workplace is near the bookstore.
Both indefinite: Vinnustaður er nálægt bókabúð = a workplace is near a bookstore.
Note that bare bókabúð is also the dative singular (indefinite). Indefinite subjects like this are less common unless you mean it generically or out of context.

What are the genders and useful case forms for these two nouns?

vinnustaður (masc.): nom sg vinnustaður, acc sg vinnustað, dat sg vinnustað, gen sg vinnustaðar. Definite: nom vinnustaðurinn, acc vinnustaðinn, dat vinnustaðnum, gen vinnustaðarins.
bókabúð (fem.): nom/acc/dat sg bókabúð, gen sg bókabúðar. Definite: nom bókabúðin, acc bókabúðina, dat bókabúðinni, gen bókabúðarinnar.

Can I front the place phrase and say Nálægt bókabúðinni er vinnustaðurinn?

Yes. Icelandic often fronts adverbials for emphasis or topic-setting. Both orders are fine; fronting puts a bit more focus on the location.

How do I say my workplace is near the bookstore?

Use a possessive after the noun: Vinnustaðurinn minn er nálægt bókabúðinni.
Without the article you get Vinnustaður minn, which means a/one workplace of mine and has a different nuance.

What’s the difference between nálægt, nærri, við hliðina á, hjá, and similar expressions?

nálægt + dat: near, close to (neutral, common).
nærri + dat: near, close to (very similar; slightly bookish in some registers).
rétt hjá + dat / hjá + dat: (right) by, at the place of.
við hliðina á + dat: next to, beside.
í nágrenni við + acc: in the vicinity of.
skammt frá + dat: a short distance from.
Choose based on how close you mean and keep the cases as shown.

How do I negate the sentence?

Place ekki after the verb: Vinnustaðurinn er ekki nálægt bókabúðinni.

How do I make it plural?

Subject plural: Vinnustaðirnir eru nálægt bókabúðinni = the workplaces are near the bookstore.
Object plural: Vinnustaðurinn er nálægt bókabúðunum = the workplace is near the bookstores.
Both plural: Vinnustaðirnir eru nálægt bókabúðunum.
Note the dative plural definite -unum in bókabúðunum.

How are the compounds vinnustaður and bókabúð built?

vinnustaður = vinna (work) in genitive singular vinnu
- staður (place) → place of work.
bókabúð = bók (book) in genitive plural bóka
- búð (shop) → shop of books.
  Using a genitive linking form (sometimes singular, sometimes plural) is the standard way to form Icelandic compounds.

Why does bókabúðinni have a double n?

Because the feminine dative singular definite ending is -inni. The stem is bókabúð; the nn comes from the ending.

How should I pronounce the tricky letters and stress?

Primary stress on the first syllable of each content word: VINNu‑…, NÁ‑lægt, BÓ‑ka‑…
ð = th in English this.
á ≈ the vowel in how.
æ ≈ the vowel in my.
ú ≈ oo in food.

Could I say Vinnustaðurinn er bókabúðinni nálægur instead?

Yes. That uses the adjective nálægur (agreeing with the subject) and still takes the dative bókabúðinni. It’s correct but sounds more formal; … er nálægt … is the most neutral form.

Why isn’t there another preposition like á or við before bókabúðinni?

Because nálægt itself is the preposition and directly takes a dative complement. Adding á or við would be redundant and ungrammatical here.