Learn Croatian: Boji li se on starosti? - Is he afraid of old age?

Starosti is the genitive singular of starost (old age). Because starost is a feminine noun that ends in a consonant, it belongs to the i-declension. For these nouns, the genitive, dative, and locative singular all take the ending -i. We use the genitive case here because the verb bojati se (to be afraid) always requires its object to be in the genitive.

How can I tell that starost is a feminine noun if it ends in a consonant?

Starost is formed by adding the suffix -ost to the adjective star (old), which works much like adding "-ness" in English. In Croatian, all abstract nouns ending in -ost (like mladost or budućnost) are feminine nouns, and they all follow this special i-declension pattern.

Since the English translation says "afraid of", why isn't there a preposition like od before starosti?

In Croatian, the verb bojati se takes the genitive case directly, without any prepositions. You don't use od here; the meaning of "of" is essentially built into the genitive ending of the noun itself.

Why is the subject on placed after li and se?