In at least two previous episodes, Beckett has stated her badge number was 41319. In this one, after her radiation detector has gone off, Beckett calls it in over the radio and quotes her badge number as 0334.
When Castle and Beckett first discover the bomb in the back of the van, the timer reads 13:21:04 and counting down just before the bombers start shooting. Minutes later, after Castle and Beckett get trapped and the bombers prepare to drive away, the timer reads 13:21:35 and counting.
A major plot point of this episode has to do with a so-called cobalt-60 "dirty bomb." Castle and Beckett are supposedly exposed to high levels of cobalt-60 radiation. Aside from the fact that no cobalt bombs have ever been built, the cobalt "tamper," as the shell of material surrounding the nuclear core of the weapon is called, is not radioactive before the explosion. The explosion transmutes the ordinary, inert cobalt-59 into its highly radioactive isotope, cobalt-60. Thus, no one could possibly be exposed to cobalt-60 radiation before the bomb goes off, unless the cobalt had been bombarded beforehand by a particle accelerator, a pointless exercise, as cobalt-60 rapidly loses radioactive effectiveness after transmutation, with a half-life of less than 5-and-a-half years.
When someone becomes an NYPD "Police Officer" they receive a badge and a badge number. That number is retained. If they get a gold badge, that is a new number. So an officer can use either their original badge number as an ordinary officer or any subsequently awarded badge number, depending on the priority they want to invoke.