- Shortest route from Chicago to Rome would be west->east, not east->west.
- 1930s air travel/mail delivery would likely have been by DC-3 (introduced 1935). Early models had a range of 1000 miles, later extended in the DC-3A to 2100 miles. Military C47 A/B aircraft had a 1600 mile range. So you'd have to introduce a few more hops on the map.
- A package would more likely have traveled via ground route, mostly ship. A trans-Asiatic route would be highly unfeasible (or would make for another Indian Jones movie: "Delivery of the Misrouted Journal").
Still, nice touch.
If you look at the history of Pan Am, it started out operating two flying boats: Sikorsky S-38 and S-40. The S-40 had a range of 875 miles (but can land in more places than a DC-3).
http://en.wikipedia.org/wiki/Pan_am#Clipper_era
That said -- Chicago to Italy should definitely be easterly.