You can remove the alternator from the car and it will start if the batttery has enough charge.
How long the engine will run on just battery is a big question mark. There are many variables, such as the age of the battery, the current state of charge, and the drain on the battery from supplying power for the ignition, fuel system, computer and accessories.
A failing battery will not provide the power needed to energize the alternator field windings. This will prevent a good alternator from making sufficent power to supply all the needs. The best test is the free one provided by your local auto parts store with their charging system testers. You can pull the alternator and have it tested separately, or the best way to do it is to drive up & have them test the whole system.