I would be very afraid to develop on 3.8.1 and try to run the same code on 3.5.1. Things have changed. 3.5.1 is ancient by web app standards. I can’t say that I have ever tried to “backport” like that, but it does worry me. Much of your code would work. WordPress itself is pathologically backwards compatible, but if your code uses new features (intentionally or not) you will have trouble.
You can install an old WordPress on the development machine, but 3.5.1 is ancient. It is unwise to run that on a production machine at all, much less to develop a site on it.
Additionally, with 3.5.1 you are likely to have numerous plugin and theme compatibility issues.
If Network Solutions won’t update to a more current WordPress, and won’t let you update, the smart move is to not use Network Solutions.
If that isn’t possible, I would install the legacy WordPress on your development server and build to that.