My first nice roadbike was a Bianchi Nuovo Record that I bought in the mid-1980s. It developed a little rust over the years, primarily because I didn't clean it enough. I sold it a few years ago and still regret it.
My advice would be to clean the frame thoroughly and sand off the paint at any rusty spots, making sure that your remove all of the rust. Be meticulous about this. Then get the frame repainted in Celeste or the original color. Bianchis are made to be Celeste, and your bike will command a higher resale value if painted that color. Trust me on this. If you just don't like Celeste, other classic colors for Bianchis were black, white and light blue (although usually with some Celeste details such as cable housings or decals).
Powder coating your frame will be more durable and less expensive, but might be harder to match colors. Your frame looks like a great starting point for a SS/fixe. Good luck with it. I wish I had hung onto my old Bianchi and converted it.