Jon, If the lathe is going "in the old outhouse", DO safeguard against rust!
Would suggest:
Vents at high and low level, can be quite small, as long as weatherproof.
NO combustion heaters to produce water vapour
If need be, be prepared to run a low power electric heater, (60 watts will keep a shop reasonably dry, although may take a couple of days to take effect. If the outhouse is larger than say 10 feet by 8 feet, (or very high)you may need more than one heater, or a higher powered one.
If you can insulate the walls and ceiling, this will help. The object is to avoid sudden changes of temperature, and to keep the temperature above the dew point.
(In a former workshop, everytime that I went in, the copious oil on the Myford had gone grey to match the paint!)
Enjoy yourself, you have the means of tackling a host of jobs. You'll wonder how you managed before!
Howard