How to fix the metal part of socket?
That can't be fixed, but it can be replaced.
That ring snapped off the piece in the right side of your photo, it connects the threaded metal on the base of the bulb to one of the wire leads.  There's no practical and reliable way to repair that thin formed sheet metal.  Don't be fooled if you assemble it and it kind of works if you wiggle the bulb.  The loose connection would be dangerous.
But you can probably find a replacement socket assembly that will work.  Lamp and fixture repair parts are available at big box and hardware stores.  Sockets, pull chain switches, etc., are mostly generic - you don't have to match the original make and model like you do with auto parts.  You just have to find one that's the same size and has the same attachment to the fixture, often just spring clips.
In the future, you can buy silicone lubricant made just for bulbs / sockets, a very light coating of that stuff will prevent this from happening.  Just be careful not to get it on the glass or plastic, your fingers, your clothes, etc., it stains.  Just a little bit on the threads of the socket or on the threads of the base of the bulb.