A bottle of Port can last up to a few weeks with a stopper to help keep the air out. If using a driven cork, the Port bottle may only retain its flavor for a few days.
Port is a red wine with a neutral grape spirit added during the fermentation process to halt it. This keeps the wine sweet and it is often served as a dessert wine.
There are many types of Port wine, but most will last for a few weeks in the refrigerator with a stopper that has a tight seal. Some bottles of Port come with their own T-stopper cork that keeps the air out of the bottle. A bottle that just has a driven cork allows more air into the bottle, hence why it will likely only last a few days.
As a rule of thumb, the older the Port wine, the faster it should be consumed. Bottles that are less than 5 years old can last for up to 5 days after opening. Vintage bottles that are up to 15 years old last about 3 days, while a 25 year old or more bottle of Port keeps best for only 48 hours. Of course, with a good stopper that time period may be lengthened a bit.