However, I saw some projects that used similar mechanism but horizontally not vertically. Any ideas on how to modify them??
Just turn it 90°?
You have to plan out your project a little before asking all your questions. Make drawings, calculate power needed etc.
The horizontal movement in the video is more or less the mechanics from a scanner or printer, Find one in a dumpster and strip it for parts (motor, axles, gears/timing belts etc.) and use the parts for your 'bot.
You keep on with the 1m height, but you don't say anything on how high it is allowed to be when retracted.
Plan, plan, plan - use plenty of pencil and paper and don't be afraid to wear down a couple of math calculators - that will make some of the answers obvious
When you come back with questions that stems from a planned out robot, it will be easier to help you and as long as you just ask about stuff that you only loosely imagined, your answers will, at best, be only loosely applicable.