Yeah, the needle bearing is the hardest to get out. Some people try to knock them out without taking that keeper ring out. To get the keeper ring out, use a small flat screwdriver or pick and stick through the small hole in the case where the bearing goes and turn the keeper ring until the ends are visible in the hole. Then, just grab one end of the keeper ring with a pair of needle nose pliers and pull the ring out through that hole. Once you get the keeper ring out, put the case in an oven and slowly bring the temperature up to about 175 degrees F. Honda makes a special tool to remove the bearing, but I've never had to use it. I just take a punch and put it through the small hole in the case where the needle bearing is and angle the punch and knock the bearing out like that.
Before you put the case in the oven, make sure all the bearings and seals are removed and make sure the gear oil fill cap is removed.
If you have a Honda factory service manual for the Fourtrax 300, it will tell you that the needle bearing can't be removed and that you will have to have a new case, but that's wrong. I've replaced a lot of them.