Do I need a shock-proof watch for cycling?
There are no special requirements for a watch for cycling. A cheap watch from the supermarkt will do as well as a decent quality watch or a sports watch (besides sport-specific functions).

Rain or road buzz might damage rather sensitive watches. Fixing a defect, loading the bike, or a crash also gives opportunities to damage a watch. Common sense will suffice to know when to take your horological masterpiece or leave it at home. In other words, there's a difference between riding to an opera and doing sports.

Most mechanical watches, for example a Seiko or something with an ETA movement, shouldn't be troubled by road buzz or the occasional cobble stretch. Please check reasonable voices in a watch forum.

Much sports riding is on a different level of impact, vibration, and risk of damage. In other words, mechanical watches are perhaps not the best choice for downhill MTB. A decent quartz watch or a G-Shock (analogue or digital doesn't matter) are preferable. Still, a dollar store digital will perform as well as the G.