Just as in a home in the country, there are a few “hidden” sources of water in an apartment. First, the moment you discover the potential for loss of water pressure, fill your bathtub. Use only the cold water tap so you don’t drain your water heater. You could go a step further and get a special large bladder you can fill and leave in the bathtub. The advantage of this is it will prevent dust and bugs from landing in what would otherwise be open water. In a pinch, use the shower curtain liner to drape over the filled tub. Either way, you can store about 60 or so gallons this way.
Go downstairs and see if you have access to the water heater. If so, buy a cheap garden hose to attach to the drain AFTER water becomes an issue. Depending on the size of your water heater, this will give you maybe 30 or so gallons.
Obviously, it rains in the city just like it does out in the sticks. The trick is to figure out how to capture it. A little ingenuity with some plastic gutters coupled with a rain barrel and you can be in business collecting rainwater on your porch/patio.
Many older buildings might still have large tanks of water on the roof. It may also be possible to drain water sitting in the sprinkler systems of buildings, but this is ONLY FOR EXTREME SITUATIONS.
No matter where you live, you should have several different methods to filter and purify water on hand at all times.