You could use WP and all it’s associated goodness for a mobile app backend (I’m going to do it soon myself).
Custom post types are not a problem. Make a page, grab the data with a WP_Query and serve up the query results as JSON instead of HTML, though you’ll have to do this bit yourself I’d wager (I’m planning on doing it).
That’s only one part of it of course, if you can give a bit more information about the project you might get some more insightful answers!