Learn Hardcore Icelandic: Ég þarf áætlun.

Here áætlun is in the accusative case as the direct object of þarf (“need”). In the singular, feminine nouns of this type have the same form in nominative, accusative and dative, so you can’t see a change in the ending.

Why is there no separate word for “a” before áætlun?

Icelandic has no independent indefinite article. Indefinite singular common nouns simply appear without any article (just like áætlun = “a plan”). The definite article is expressed by a suffix (see next question).

How would I say “I need the plan” instead of “a plan”?

Attach the definite suffix –in to the noun.
“Ég þarf áætlunina.”
Here áætlunina = “the plan” (accusative feminine singular definite).

How do I form a yes/no question “Do I need a plan?”

Invert the verb and subject:
“Þarf ég áætlun?”
(Verb-first word order signals a yes/no question.)

How do I say “I don’t need a plan”?

Place the negation ekki after the verb:
“Ég þarf ekki áætlun.”

What is the plural of áætlun, and how would I say “I need plans”?

The plural nominative/accusative is áætlanir.
“I need plans” → “Ég þarf áætlanir.”

How do I add an adjective, for example “I need a detailed plan”?

Adjectives must agree in case, gender and number. Here you need the accusative feminine singular form of nákvæmur (“detailed”), which is nákvæma:
“Ég þarf nákvæma áætlun.”

How would I express past or future need instead of present?

Past: change þarf to þurfti (preterite) → “Ég þurfti áætlun” (“I needed a plan”).
Future: use the auxiliary mun + infinitive → “Ég mun þurfa áætlun” (“I will need a plan”).