No, time does not become irrelevant. It's perfectly normal for things inside the black hole. Here's the space time diagram for our universe on the right, and a black hole at the top-left. Time is the vertical axis, space is the horizontal. The speed of light is a 45° angle, and the solid lines are event horizons. The hourglass shapes are the cones of all your possible futures and pasts (aka, anywhere that isn't faster than the speed of light from a position). Notice the space-time diagram looks exactly the same on the other side of the horizon. To get back through though you'd have to travel faster than that 45° angle, which is impossible.

Edit: I remembered there's a PBS Space Time video that will help you understand this if you don't. It goes a lot further than just this version of the diagram.