With all the horrors that have unfolded in the Middle East since just last year, the Arab Spring — the pro-democracy uprising that began with protests in Tunisia and Egypt in 2010, leading to the dissolution of authoritarian regimes — can seem like a glimmer from a halcyon era. Yet as a new musical, “We Live in Cairo,” reminds us, that era was itself short-lived, and produced its own bloodshed and despair.