If it is just where the chain rubs the sprocket teeth then it should be good with fresh oil on the chain. When you ride the surface rust will be worn off and then covered with the new oil.

The old Schwinn chainrings seemed to be chrome plated better than nowadays, however, with environmental restrictions, it may not actually be chrome anymore. Or as resistant or thick as it used to be.

Let us know how it looks after a ride or two.