You can store uncorked white wine, usually up to five days if you keep it in a fridge. You may use your regular refrigerator or a wine fridge. A regular fridge should not be set at its coolest because the temperature will drop below six degrees Celsius and that is not good for whites. A wine fridge will cost you a few hundred dollars but it is worthwhile if you are a connoisseur.
You must reinsert the cork into the bottle. Keep the clean side facing up and insert the stained side. The clean side is actually unclean because it has been exposed to dust and dirt, possibly contaminants too. The wine will go bad when it comes in contact with the clean side of the cork. You have to store white wine sideways to keep the cork moist. This is true for unopened and opened bottles. There is no way you can avoid the cork being in contact with the wine. You should not store an opened bottle for too long if you have not deoxygenated the vacant space.
There are ways to suck out the oxygen from the bottle. Whether you have had just one glass or several, the space inside the bottle will contain air and hence oxygen. This will interact with the wine and it shall go bad sooner than you think. You can get devices to extract the oxygen out or to pump in a gas that will coat the surface of the wine inside the bottle and prevent it from interacting with the air. Such options will give you a few more days than usual. You should be able to store uncorked white wine for up to seven days in a fridge if you get rid of the air. You may also use a smaller bottle if you have one so the amount of air inside is reduced.