No.
And yes.
Physically the pinouts are the same for any lga115x series, so any lga115x cooler will physically work in that respect.
No because you are moving from a 65w 4c/4t cpu to a 95w 6c/6t cpu and that's just TDP. Expect actual heat outputs to be quite different under gaming loads.
You can wait 2 days, it's not going to kill you, but there's no point pushing your luck.
Of course that means nothing if your current cooler is considerably better than the original stock cooler that came with the 6400, in which case even a budget cooler like a hyper212 is good for the 9700k, just don't try OC yet.