At barbecue joints, coffee counters and bottle-service nightclubs, a coming clampdown on big, sugary soft drinks is beginning to take shape on tables and menus in a city that thrives on eating and going out.

Some restaurants are ordering smaller glasses. Dunkin’ Donuts shops are telling customers they’ll have to sweeten and flavor their own coffee. Coca-Cola has printed posters explaining the new rules, and a bowling lounge is squeezing carrot and beet juice as a potential substitute for pitchers of soda at family parties — all in preparation for the nation’s first limit on the size of sugar-laden beverages, set to take effect Tuesday.

Some businesses are holding off, hoping a court challenge nixes or at least delays the restriction. But many are getting ready for tasks including reprinting menus and changing movie theaters’ supersized soda-and-popcorn deals.