Learn Hardcore Icelandic: Ég drekk yfirleitt bara einn bolla af kaffi.

bara: very common, informal, also “just/simply”
aðeins: neutral to slightly formal; can also mean “slightly”
einungis: formal/literary All three work here.

It’s accusative singular. The verb drekka takes an accusative object, so masculine weak noun bolli becomes bolla in the accusative. With the numeral, the noun still follows the case required by the verb: (ég) drekk einn bolla.

Why is it af kaffi? Which case does af take?

Af governs the dative. Kaffi (neuter) has the same form in nominative and dative singular, so you see no change. The pattern for measures is container + af + dative:

bolli af kaffi (cup of coffee)
glas af víni (glass of wine)
skeið af sykri (spoon of sugar)

Could I say einn kaffibolla instead of einn bolla af kaffi?

Yes, very common. Kaffibolli is a compound and usually means “a cup of coffee” (the drink).

einn bolla af kaffi = emphasizes the measure “a cup of coffee (as a quantity)”
einn kaffibolla = the standard everyday way to refer to one cup of coffee

Is einn here the numeral “one,” or is it acting like the English article “a”?