There are AT and ATX power supplies. Assuming your emachine has a 250w ATX and your friend will give you the 350w ATX power supply, sure you should switch for safety sake; 250w is too low for a R9800 pro, or very very close anyways depending on the rest of your system.
The P4 power supply have 2 extra power cables, one of which you would use on an NF2 Athlon XP system also. Most Power supplies will have these and it is no harm to have them and not use them. So if you have an Athlon don't worry about putting in a P4 ready power supply.
I doubt either your 250w or the 350w are dual fan ones, but typically for Athlon XP's the better cooling ones have a fan on the bottom of the powersupply as well as in the back. This provides better cooling for those systems. Don't worry if both of your's don't have this. If the emachine does have this, but the 350w doesn't, As long as you have a rear exhaust fan under the power supply you should still be fine. If the emachine uses the power supply as the only exhaust, then you may need to add a rear case fan or rear slot fan for lowering the case temps.
Only things I'd check is that the exhuast fans in both power supply blow air in the same direction. Most pull air out of the case and blow the air out the back of the power supply. Older ones sometimes did the opposite in some oem systems. I doubt that is the case, but if your emachines for some reason sucks air in through the power supply and your swap in a power supply that does the opposite, you may kill the airflow design of the system. But this is a slim chance as even emachines probably has the PS fan blowing out of the case.
Hope this isn't confusing. Basically I say yes grab your friends 350w if both are ATX. Since I can't see your system, I am just trying to offer a little more info to keep you out of possible trouble.
ABIT IS7, P4 2.6C, 512MB Corsair TwinX PC3200LL, Radeon 9800 Pro, Santa Cruz, TruePower 430watt