Or did anyone at that period? My understanding is he was popular with average Romans, so I'm wondering if he would have been honored in some way (by his soldiers perhaps?) or just dumped in an unmarked grave or something worse.
During this period Cremation was the most common practice for the deceased. Both Julius Caesar and Augustus were cremated after their deaths.
In Julius Caesar's case, the story that comes to us from Suetonius and Appian, among others is this: After the assassination Marc Antony gave a eulogy of Caesar from the Rostrum, the speakers platform, in the Roman Forum. Reminding the people and veterans of all that Caesar had done for them and showing off the bloody toga, The people then took the body of Caesar, which had been intended to be burned and the interred elsewhere, and made a pyre in the forum before the Rostrum and burned it there. We don't know where Caesar's ashes were later interred, unfortunately, though in Suetonius account Caesar's funeral was originally to be held in the Campus Martius (which was outside the pomerium and the Romans usually interred the dead outside of cities) beside the "Tomb of Julia." So I think it is safe to assume, but not absolutely certain, that he was interred in a family tomb for the Julii. Since Caesar was deified as Divus Julius after his death, there was a temple erected to him in the forum, and according to tradition the site where he was cremated by the mob remains in the niche in from of the Rostum that comprised the front porch of the Temple, and which is still visible in Rome today.
In Augustus case he died in the town of Nola, in near Naples. A funeral procession moved north from Nola to Rome, where Augustus had built a mausoleum for himself and his family. The mausoleum, also in the Campus Martius, also can still be seen and visited in Rome today. It was outside the Mausoleum where Tiberius delivered the eulogy and where the body was cremated, before being interred within.