Rust on Amsonia
Yes, this is a type of rust. You need to cut down and remove all infected plants so that it does not overwinter in your garden. You can do that now or, if it has a lot of foliage that is still healthy green and feeding the roots, you may want to wait until it has gone dormant this fall. Either way, get that infected plant material out of there.
No, it will not spread to other plants.
Here is a good link to information on preventing it in the future: http://www.backyardgardener.com/plantname/pd_ebab.html