Is that possible to pause a system?

The alarm flag is to be set if fifo gets empty or full. During system development you will then need to readjust the operating point or fifo size until the alarm disappears but keep it in the delivered design so that your manager wouldn't sack you in case the design fails in field.

The processing behind fifo will continue nonstop and there is no need to pause the flow.

If your flow is packet based then you need to be careful not to break packet boundary in the fifo. Some system allow inserting null data in case it is needed when flow is too slow.