Learn Hardcore Icelandic: Ég fæ mér kaldan drykk eftir vinnu.

Literally ég = “I,” fæ = “get” (1st sg. present of fá), mér = “to myself” (dative). Idiomatically it means “I’ll get myself” (e.g. “I’ll grab myself”).

Why is mér in the dative case? Why not the accusative?

The verb fá requires a dative pronoun when it’s used reflexively as fá sér (“get for oneself”). For 1st person singular the reflexive pronoun is mér in dative, not accusative.

Why is kaldan drykk in the accusative and not kaldur drykkur or kalt drykk?

kaldur drykkur is nominative (“a cold drink” as subject). Here it’s the direct object, so both adjective and noun go to masculine singular accusative:
kaldur → kaldan
drykkur → drykk
Hence kaldan drykk.

Could you say kalt drykk instead of kaldan drykk?

No. kalt is the neuter form of the adjective. For a masculine noun like drykkur, you must use the masculine accusative form kaldan.

Why is it eftir vinnu and not eftir vinnuna?

The preposition eftir governs the dative. vinnu is the dative singular of vinna used indefinitely (“work” in general). If you said vinnuna, you’d be referring to a specific piece of work (“the work”).

Can the present tense fæ refer to the future here?

Yes. In Icelandic, the present tense often describes a planned or near-future action. So Ég fæ mér kaldan drykk eftir vinnu can be understood as “I’ll get myself a cold drink after work.”

Is fá mér the same as kaupa mér or taka mér?

Not exactly. fá mér = “get for myself” (may include buying, ordering, being offered, etc.). kaupa mér = specifically “buy for myself.” taka mér = “take for myself,” more like helping yourself if something is freely available.

Could you add fer to emphasize going somewhere, as in Ég fer og fæ mér…?

Yes. Ég fer og fæ mér kaldan drykk eftir vinnu (“I’ll go and get myself…”). It’s a common colloquial pattern: fer (“go”) + og + verb (“do something”).