Learn French: Tu aurais dû redémarrer l'ordinateur avant la panne ! - You should have restarted the computer before the outage !

Tu as failli redémarrer means "You almost restarted" (a near miss where you were on the verge of doing it, but didn't). In contrast, this sentence uses tu aurais dû redémarrer, which means "You should have restarted" (expressing a reproach or regret about a duty you failed to perform). Both verbs are followed directly by an infinitive, but faillir takes the passé composé for past events, whereas expressing "should have" requires the past conditional of devoir.

How exactly is "should have restarted" constructed grammatically here?

It is built using the past conditional (conditionnel passé) of the verb devoir (to have to / must), followed by an infinitive. You conjugate the auxiliary avoir in the present conditional (tu aurais), add the past participle of devoir (dû), and then attach the action verb in its infinitive form (redémarrer).

Why does the past participle dû have a circumflex accent?

The circumflex accent (the little hat) is used purely to distinguish this past participle from the partitive article du (meaning "some", as in du pain). Keep in mind that this accent disappears if the past participle takes an agreement, becoming due in the feminine and dus or dues in the plural.

What kinds of outages or breakdowns does cover?