If I use primer on a rusty hinge, can I paint it and get away without sanding it first?
Depends on if you want it to look good or not. If you don't care just go to local big box and grab a can of spray paint - I use the rustoleum auto body primer for any metals. Then I go over it will whatever type of spray paint - usually a bronze or satin finish. If you don't sand it you will have rust bubbles in your paint and unless you use an obnoxious amount of paint, you will have pitting and bursted bubbles, given that these plates are hitting each other constantly.
It is really all about the look you want. I have seen people keep rust for an antique look and just a light spray of metal paint. And then I have seen people make them look like new which involves sanding and painting as I prescribed above.
You "can", but odds are it'll look terrible and likely as the rust flakes the paint will come off with it.
you're better off cleaning this and painting it properly.
If you want to avoid sanding because it's antique you can try electrolysis to attempt to reverse the rusting process. If you don't want to sand because it's annoying... yeah... just fix it right.