Yes a car should run without the battery hooked up after you get it started.
Of course you need the battery to supply enough voltage to the cars engine to get it started and to turn over the engine with the starter.
But after that the engine should run just fine without the battery as the alternator spinning should create enough voltage to keep the engine running.
My dad told me of a test for an alternator that if you start the engine and then unhook the battery positive side the engine should still run if the alternator is good.
And if the engine dies then the alternator is bad.
I have unhooked a battery before on a car to see if it would stay running without the battery hooked up and it did run.
But you shouldn't drive a car without the battery because it could cause other issues with the cars electrical system.