Learn Italian: Lui è disposto ad apparecchiare da solo? - Is he willing to set the table alone?

The adjective disposto (willing) requires the preposition a when followed by an infinitive verb. It becomes ad here because the next word, apparecchiare, starts with the same vowel 'a', and the extra 'd' makes it easier to say.

Is there a trick to knowing which adjectives take "a" versus "di" before an infinitive?

There is no universal rule, so you often have to learn them as set phrases. However, adjectives that express readiness, a tendency, or a willingness to do something (like disposto, pronto, or abituato) usually take the preposition a.

Is the 'd' in "ad" mandatory before any word starting with a vowel?

Generally, in modern Italian, the euphonic 'd' is only added to a (becoming ad) or e (becoming ed) when the following word begins with the exact same vowel. Since apparecchiare starts with an 'a', using ad is standard here to avoid two 'a' sounds blending awkwardly.

Why does the sentence use "da solo" instead of just "solo"?

The phrase da solo specifically means 'by himself' or 'on his own', emphasizing that he is performing the action without anyone's help. If you just used the adjective solo, it would mean he is physically alone or lonely.