Do I need to go to the doctor if I was shaving my "down stairs" with a razor that had a little spot of rust?

No. Touching rust if not a problem. People use, handle, and play with many metal objects or tools that have rust on them. The "stepping on a rusty nail" concern occurs because germs are pushed through the skin wound, deep enough to cause an infection. However, razors shave better if they are smooth and rust-free, so a rusty razor should be safely disposed of.
Rusty razor. Were you cut by the blade? How long ago was your last tetanus shot? If it has been ten or more years ago - would get another one.