Sean,
The 250/350cc Aero models are not reputed to be very powerful. "Wouldn't pull the skin off a rice pudding" is the common description. There seem to be a fair number of engines that have been separated from the frames. They share the same frame and forks as the 600cc model, and the engine weighs pretty much the same. The 600 is no ball of fire either, but it is very tractable. It trundles up to about 62mph and then plateaus. A 350 would be slower yet. They do handle very pleasantly; the frame is quite light and all the weight is down low. Stopping is the typical Douglas band brake gamble; you could have quite good brakes (for the period) or appallingly bad.
As Aero36 mentioned, unless you are going to start it up and allow it to get to operating temperature, you are better off not starting it at all. If it is going to be parked up for a long time, it would be a good idea to drain off the fuel and run the float bowl dry. Particularly if you live in an area with ethanol in the fuel.
It would help to know what type of oil it has been run on in the past. Most oils now, even mono-grades, have some detergent additives. If the bike has been run in recent times it probably already has been run on detergent oil. But if it has never seen detergent oil before, switching over can release a deluge of previously undisturbed sludge to circulate around the engine. Most folk will not go to the trouble and expense to strip an engine to clean out the sludge traps before switching over to modern oil, so if it has not been done you might want to run a oil flushing regimen. Since these old bikes do not have any form of oil filtration, the detergent oil keeps the wear partials and contaminants in suspension. Another reason more frequent oil changes are recommended when running old engines on modern oils. The only way to get rid of the contaminants is to change the oil.
-Doug
