well blinking eyes are a real challenge even for characters with out such a extreme eye shape, mouth sync is somthing different ,
but recardless you are amking a assumption that many novices and people with little puppet designing, and building expierience make, that is thinking of the puppet as a three dimensional cartoon ,
cartoons can be designed to be almost anything, and still move in a fluid manner that is believable, puppets , hand puppets especially need to work within the confines of the medium, even with all the video tricks, and mechanisims, out thier , thier are somethinsg a puppet cannot dop in real time,
look closley at the design you are trying to owrk around, he has HUGE oval eyes, and a very skinny neck, a long body and arms , now how will u make all those things work as a three D creation, with out losing all of the trademark featre that make the design work?
aside for the fact that you will never achiev a proper blink with those oval eyes in the confines of that small head, i suggest you start with some simpler ideas, and learn a little more before tackeling aproject that will never reveal the results you want