[masukomi]

not aging well implies the thing gets worse the older it is. So, a more accurate interpretation would be "bad news gets worse the older it is" Although that seems unlikely to be what was intended.

[gameman144]

I read it the same way, with the sentiment of "People will be more upset if you delay telling them bad news."

e.g.

"We got a letter for past-due rent yesterday, and I'm telling you now"

vs.

"We've been getting letters for past-due rent the last few months, and I'm telling you now."

[Konohamaru]

"not age well" in food means "expires quickly." Milk does not age well because it quickly expires. So "bad news does not age well" means that bad news quickly expires. :)