I evaluated the free WP RSS Aggregator plugin and the paid extension, Feed to Post. Using both, I was able to import new posts with a custom post from my feed. I was able to assign terms in a custom taxonomy and a tag to the newly created posts.
Regrettably, the RSS feed I need to import does not validate, and it fails in such a way as to make the Feed to Post extension fail to import new items in the feed. Even though the problem was with my feed and not the plugin, I still got great response and service from the developers. Because my feed was so badly malformed as to not really qualify as an RSS feed, I ended up using a different plugin to import posts. Still, I think this plugin does handle valid RSS feeds perfectly well. And the support I got was great.
I recommend validating the RSS feeds you want to import prior to attempting to use this plugin to import them, and definitely prior purchasing any of the extensions. But if you have a valid feed, I'm confident this plugin will import it for you, and I believe you'll get great support should you need it.