Opened bottles of Prosecco or other sparkling wines will start to lose their carbonation and flavor after about 1-3 days, even when stored in the refrigerator with a wine stopper. If you use a special sparkling wine stopper that seals the bottle airtight, you may be able to extend this period to around 3-5 days, but the quality will still gradually decline after opening.
Remember to store the bottle upright to minimize the wine's exposure to oxygen. The key to preserving wine, especially sparkling wine, is to limit its exposure to oxygen which causes the wine to oxidize and lose its flavor.
It's worth mentioning that although the flavor will degrade over time, it doesn't mean that the wine will be unsafe to drink. It might just not taste as good. It's best to use your judgment, if it tastes off, it's probably past its best.