Learn Russian: Какого числа приезд директора? — Двадцатого августа. - On what date is the director's arrival?

In Russian, to express that an event happens on a specific date, you use the genitive case without any prepositions. So to ask about a date, you literally ask 'Of what date?' (Какого числа). The answer follows the exact same pattern: Двадцатого августа (literally 'Of the twentieth of August').

In Двадцатого августа, why does the number have a neuter ending instead of matching the masculine month (август)?

The ordinal number двадцатого (the twentieth) does not describe the month. Instead, it describes the hidden word число (date or number), which is a neuter noun. Because we need the genitive case to say 'on the twentieth', the neuter word takes the -ого ending.

Why is директора in the genitive case?

Here, the genitive case is used to show possession, just like adding an apostrophe-s or 'of' in English. Since приезд means 'arrival', приезд директора translates to 'the arrival of the director' or 'the director's arrival'.

Where is the verb in the first sentence?

There is no verb because Russian normally omits the present tense verb 'to be' (is/are). The literal structure here is 'Of what date [is] the arrival of the director?'.