Yes, I think NPS is a fairly good indicator, but it’s a “one-dimensional” indicator. No matter how many customers you survey it is still just a survey, without much context.
Some other ways you can get a sense of how customers think of (and just as important, use) your site or product:
Analytics can be a useful indicator, but not the way you might think. If you have any way of tracking through analytics how often the application is used it can give you some great insights into how your product or site is used. If customers tend to “go” to a particular “place” often this will tell you where to focus your attention. It also may help you identify potential pain points in your experience (I’ve used analytics to identify “drop off” points in an application process to make hat process better).
Social reputation is also a key indicator. Set up saved searches for your product name(s) in Facebook, Twitter and Google+ to monitor what customers are saying about your offerings. While not “scientific,” it can give you some practical “real world” understanding.
Finally, have regular in depth discussions with customers. Even if it’s a group of 10 people, having a conversation where they discuss what they like and don’t like about your site or application will give you insight into potential areas of improvement.