I'd challenge both claims that he was considered the best emperor and that he began Rome's decline.

Firstly, throughout most of antiquity Trajan was considered to be the most popular emporer. He was supported by the people, the senate and the military throughout his reign. He expanded the empire, defeated the Dacians and ended the Jewish revolt. He was so popular that a Roman pope claimed that God resurrected Trajan and allowed the Pope to baptise him in order to admit him to heaven, since the idea of such a popular pagan emperor residing in hell was so controversial.

Furthermore labeling Augustus as the start of Rome's decline is to vague to answer. What aspect of Rome's decline are you referring to: economic, military, social and moral or rather the decline of the Roman Republic. Economically and militarily, Augustus began vast improvements in both; expanding the empire, consolidating the provinces, reducing slave labour to provide work for Romans. The decline on the roman military began after Hadrian's policy of ending expansion of the empire. Economic decline is harder to pin down, one might argue it began with Nero's devaluation of the currency or perhaps later during the economic crisis faced by Diocletian. Augustus also oversaw the cultural peak of Rome, patronising some of the greatest poets of the period, notably Virgil and Horace. I won't comment on the moral state of Augustus' rome because it's to ambiguous.

As for his role in the end of the Republic, he did enact the final killing blow by beginning the imperial period. However that decline began a century before with Gracchi brothers reforms and later the 1st triumvirate. The Republic was falling already, by the time Augustus was born.

In summary, Augustus was neither a source of Rome's decline, nor was he regarded as the greatest emperor (at least not after Trajan's reign).

Apologies for lack of sourcing. On my mobile. Will fix when at my PC if there's interest.