The most common method of fighting fire in antiquity was simply to drench the fire with water carried in buckets--Juvenal amusingly pokes fun at the rich man who keeps gangs of slaves awake all night with water buckets in case his expensive property catches fire, and in general in the Republican city fighting fire was the duty of the individual and his neighbors, until the fire started getting out of hand. Under Nero it became a requirement, in fact, that every residence should maintain the necessary equestrian for fighting fires, though one wonders how realistic such an order was. Other than water we're told of rags soaked in vinegar being used to smother fires, and when all else failed the burning building could be demolished to prevent the fire from spreading--the jurists actually get into a debate over whether you can sue if a neighbor demolishes your property to prevent his from catching fire (if your property is not actually on fire--if it is and the fire is uncontrollable they all agree you have no right to sue). Pliny (and a few others--it's in the Digest) mention a sort of fire-pump that apparently had a hose of some kind, but these were apparently uncommon and probably not that effective.

Fire was a constant threat in the city. Open fires were everywhere in the city, in hearths particularly but the threat existed every time a fire was started or moved. During Crassus' lifetime none of the building codes mentioned by imperial authors and in the Digest existed--in later times private dwellings had to be a certain distance from public buildings, and starting with Nero buildings could no longer share a wall, which was common practice. In the Republic most dwellings were crowded together with shoddy construction, often just rubble faced with stone, sharing a thin common wall with the neighboring building. Streets were narrow and buildings hung menacingly over them and neighboring structures--the jurists even mention neighboring roofs catching fire and dropping embers onto your own.