They may well have had some form of very simple lathe, even if the motive power was an assistant turning a handle, or it was perhaps a pole lathe.
The basic concept of turning, in wood, seems to have been used by the Ancient Egyptians; with an assistant revolving the work-piece directly. The Romans used simple, probably pole-, lathes for furniture making.
As for tools, did the Ancient Greeks have iron? Or could they have used knapped flints in suitable holders to cut the relatively soft bronze? (Flint is silica, I think harder than tungsten-carbide!)
Artefacts from the Bronze Age often show engraved patterns and other examples of cut work, so having discovered how to make the alloys in the first place, the ancients don't seem to have taken too long (historically) to develop methods of working them into items often both functional and aesthetically beautiful.