Yeah, if he is sitting in a chair, then stands up straight very fast, he 'jumped to his feet'. (You can't jump on your own feet, but if your friend 'jumped on your feet' he might break your toes) Are you thinking about break-dancing, where someone lying prone, on his stomach, arches his back, then rocks and 'springs' or jumps to his feet? Spring is good to use if you are in a wound up position, then suddenly unwind your body.

Merry Christmas! "I'd like to ask if one person lies on his feet and jumps without standing up" ...this makes no sense whatsoever, even if you replace "feet" with "stomach". Maybe "bounce"? I'd need to see a video of what you're trying to describe. Maybe "jumps using his stomach". If it's in a nightclub dancefloor, then we say "do the worm". If the result of the jump was the person standing, then "jumps to his feet" is what we use. (Although that probably wasn't your question.)