Does the new month start at midnight or 5am?

It's midnight. I happened to be playing yesterday around midnight and I started catching April fish from midnight on.

I believe it switched over at midnight, as soon as it became April. I caught my stringfish with about fifteen minutes to spare. I checked its entry after midnight, and it showed it wasn't available during any hours and that it was currently considered April. It's hard to prove a negative, but I also didn't see any emperor butterflies after midnight. If you check their entry, I assume it looks similar to the stringfish one in terms of current hours of availability. Hope that helps.