What's the name of website testing method when the new feature is shown only for some users (not for all), for instance 20%? Are there some online services for that? I know about A/B testing but I'm not sure that's what I need.
What you're looking for is Google's Content Experiments.
You're right, this is an A/B test, most specifically a content test in order to take conclusions on what is the highest converting page.
It's possible to send just a percentage of the traffic as you request and the results are all inside Google Analytics, so you can integrate with Goals and Conversion funnels. Besides, is inside a platform you most likely already know.
You have other tutorials available, like this one, for example: http://www.newmediacampaigns.com/blog/google-analytics-ab-split-test-tutorial
You can read about experiments here: https://support.google.com/analytics/answer/1745147?topic=1745207