The short answer is yes, it will work fine(you can find both better and worse thermal paste). I agree with the poster above about very little gains.
How if you are swapping the cooler or required it off for some reason, then yes, you can use that for re installation.
The 4870 cooler also has heat transfer pads for the ram(and maybe other chips, I forget as it s been a while since I removed the cooler from a 4870), so ensure those are at least half clean if you are to have any chance to re-use then. If they are not, you should replace those as well.
Also note that high idle temps(70-80) is normal for that card since it slows the fan down lots at idle.