When you want to clean the lathe, use something like kerosene or mineral spirits and clean rags. Make sure you are working either outdoors (prior to bringing the lathe into your shop) or else make sure to have good ventilation with a positive air flow (opened windows, fans, etc). Personal protective gear is a must, something like neoprene or nitrile gloves which won't dissolve in mineral spirits or kerosene, eye protection, and make sure the air is moving from you towards the lathe so you do not inhale lungfuls of solvent vapors.
A little steel wool or "Scotchbrite" pads soaked in mineral spirits or kerosene will do wonders for the handles, handwheels, and micrometer collars. Do not use Scotchbrite pads on the bedways or any other machined surfaces. Scraping off the cosmoline (or "Rust Veto"- another popular preservative) with single-edge razor blades carefully wielded so as not to scratch or dig into the machined surfaces is one option, using a piece of brass shim stock or a piece of plastic (such as "Lexan" or acrylic) filed to a nice sharp edge is a safer bet for scraping preservative off machined surfaces.
When you have the lathe fully cleaned, I'd drain the headstock of any old oil. Then, I would recommend flushing the headstock by filling with kerosene or diesel fuel and running the lathe under power at low speeds for a few minutes. Do not forget to oil the headstock spindle bearings- this is a "plain bearing lathe" (bronze bearings), and I do not know if the headstock gearing slings oil up to the bearings once the lathe is running. I'd use an ISO 46 oil (about a 20 weight), non detergent, no additives. Tractor Hydraulic Oil is what I use in my own machine tools for well over 30 years. Tractor Hydraulic Oil is an ISO 46 oil known as a "DTE" (Dynamo, Turbine,Engine) oil, a designation that predates the automobile and is still used for industrial and powerplant lube oils.
After you run the lathe for a few minutes with the kerosene or diesel fuel in the headstock, drain it out. Open the headstock by removing the top cover. Inspect the internals for any pre-existing damage, and feel around the bottom of the headstock housing for any shrapnel or metal particulate. I go fishing with a mechanic's magnet on an extendable wand for this sort of thing. Word of caution: disconnect the power to the lathe when you are opening and inside the headstock. One bump of the reversing switch, if the power is connected, and your hands go thru the meat grinder. Open the circuit breaker that feeds the lathe at the service panel if no other means of disconnecting the power exists once you install the lathe. When you have the headstock nice and clean inside, replace the cover. A silicon 'gasket eliminator' applied in neat beads is fine. Refill the headstock with ISO 46 oil and it should be ready to go.
I'd flush the apron with kerosene or diesel fuel as well. The apron will have a plunger pump that pumps oil up to the various points like the saddle to lube the ways, cross slide, and other mechanism parts in the apron. There is a small pipe plug on the underside of the apron to drain the apron. Fill with kerosene and start working the plunger pump to flush the oil drillings. You should eventually see kerosene on the bedways after working the plunger pump for a few dozen pushes.
Drain the kerosene and refill the apron with a light way lube, or you can use the same ISO 46 oil. I use "Husqvarna Bar and Chain Oil" as it is what is locally available, I have it for my own chainsaw, and it is a light way lube.
Flush the quick change gearbox thru with kerosene as well, just pouring it into the oiling points where you swing away the top cover (there will be an instruction on the top cover saying to 'oil with the lever in this position' with an arrow).
As you clean your lathe, identify all the oiling points. Flushing the smaller ones thru using automotive brake parts cleaner (in spray cans with a thin 'straw' or 'wand') is what I do, inserting the wand in the oil holes.
A surface cleaning of the lathe is one thing, but the working parts in the headstock, quick change gearbox, apron, etc all bear some cleaning and flushing before the lathe is put into operation.