Given Nero's rather brutal reputation, what with the ordered suicide of Seneca, Christian persecution, and allegedly playing an instrument while Rome was on fire, its easy to imagine that he wasn't the most popular. But do we have any accurate sources describing how the public regarded him in the following years, and how they regarded him during his reign? As an additional question, what would everyday life for the common citizen have been like in 69 C.E, during the Year of Four Emperors?
As others have noted, the sources we rely on come from the elites of the empire and they loathed Nero - obviously we're getting a biased view of him. We know he must have had some popularity since several 'false' Nero's popped up after his death - you don't pretend to be someone's who's unpopular. But it is noteworthy that both false Nero's came from Greece which suggests greater popularity there - not surprising considering his love of things Hellenic and lifting taxes during the Isthmian Games. And following the Great Fire of 64 CE, he opened up public spaces to house the homeless and started the rebuilding of Rome. His frequent spectacles were quite popular with the people of Rome.
But if even half of the anecdotes we have of him are true then he was a pretty piss-poor emperor, especially if we compare him to Claudius who took concrete steps to improve the quality of life for the citizens of Rome (e.g. building the portus, legislation to encourage more shipping, winter sailing, etc...) Instead we see him more concerned with chasing his dream of being a chariot driver or a singer - fine if you're a private citizen but not as cool if you're the Princeps and you have other duties (part of which is not pissing off the elites you rely on for managerial tasks).
On a side note, there's apparently a single scholar put forward a theory that maybe those darned Christians did start the fire of 64 CE. He noted that it began exactly 418 years, 418 months, and 418 days after the Gallic sack of Rome in 390 BCE. Maybe a fringe group of Christians in Rome, at this point only a generation removed from the Crucifixion, got a bit radical (and not the good kind) and decided to start the end times and bring back the saviour assisted by this 'magic formula'? Or more likely the fire started by accident - as happened in the many other fires which hit Rome. But still, I'd keep an eye on those Christians, no knowing what they'll do next...
Thank you all for your answers, they've both educated me and changed by perspective on Nero as an Emperor. But addressing my second question, what would life have been like as a common person or even as a politician during the Year of Four Emperors in mainland Italy? How did the people feel about the constant change of leadership after the loss of their former leader, which we have discovered may not have been so unpopular after all?