Just a minor correction: it was Justinian's general, Belisarius, who recaptured Rome, not the Emperor himself. Justinian generally avoided "getting his hands dirty" and appears to not have been interested in being a frontline military commander. He was a very shrewd and imaginative political leader with a single, great vision: to restore the Roman Empire back to its former glory.
Anyways, you can read about the capture of Rome by Belisarius in Procopius' History of the Wars, in the section titled: The Gothic Wars. Thankfully, someone has kindly compiled this source online. The relevant sections run from Chapter XVI of Book V through Chapter XIII of Book VI.
To summarize, it seems that the while much damage had been done to the city in the vicious sacking by the Vandals in AD 455, the civic infrastructure of the city still remained for the most part, functional. Although the population was severely diminished. Comments are made in The History of the Wars regarding aquaducts, markets, and other important civilian structures. It even seems that the people of the city still retained their cultured air about them. It is mentioned, however, that the defenses needed to be reinforced and rebuilt by the soldiers of Belisarius, however.
pretty awful. The population had shrunk to a fraction of its former size, perhaps down to 50k from around one million in the Antonine period. Consequently most of the city had fallen into disrepair and it was like a graveyard of tumbledown ruins. Only the area around the Lateran Palace and st. Peter's remained in even good condition. Also the hydraulic water mills which ground the city's flour had been destroyed as had the aqueducts that brought water to the city. Thus the fountains were silent for the first time in 1000 years.