Unless the hitch has been modified, it likely has a 2" Marvel coupler and will require a 2" ball. My suggestion would be to search out one of the "heavy duty" 2" balls with the 6,000 pound weight carrying rating - - they usually must be purchased from a trailer specialty store or RV outlet as the ones typically handled by Wal Mart/K-Mart, etc usually do not have a weight carrying rating much in excess of 2,000 pounds. So far as hitch height is concerned, Airstream coaches of that era typically had a hitch height between 17.75" and 18.25" - - I checked the Airstream site, but they have only spotty coverage of such information for trailers of the 1950s.
Airstream Trailer Weights, Hitch Heights
The main idea with the hitch height is to provide a near level towing stance for the trailer - - it is extremely important for tandem axles to keep tire wear and loading uniform - - even with a single axle it is important to help insure that rear bumper and black tank valve aren't unnecessarily exposed to dragging on uneven pavement as well as to maintain proper balance for safe towing.
Quite honestly, even if you have the spring bars, it will likely be easier to buy a matched set that includes the adjustable ball mount, spring bars, and snap-up brackets - - there will likely be far fewer frustrations setting up something with all parts accounted for rather than trying to piece together the vintage pieces. This would also give you the opportunity to consider a Reese Dual Cam system which combines a weight distributing hitch along with quality sway control.
You mention needing to purchase tires. You might want to prepare yourself for the probability that you will also need to buy new wheels as well. Given the recent history of the trailer, it may very well have its original "split-rim" wheels that most tire dealers refuse to service. New trailer wheels aren't particularly exensive (the wheels that I purchased for my Overlander were $40 each when purchased with tires at my Good Year dealer). One caution is that the wheel wells on the Vintage single axle trailers can be quite tight so it is very desirable, if at all possible, to take the trailer to the tire dealer and allow him/her to measure and fit the tires/wheels - - that way, if the tires/wheels don't fit the dealer will be more likely to handle the situation without hassles.
Good luck with your new Vintage!