How to Replace Rails in a Split Rail Fence
This repair is based on Split Rail or Cedar Post and Rail fencing.
First you would have to remove the existing post by digging a rather large hole all the way around. Trying not to snap the post if it has not already broken off at the base.
Once you remove the post I would suggest digging the post hole to having a diameter of about 12 inches or so round. At the top of that hole it would be best if the hole opens up to a 20 inch diameter hole... almost like a funnel going downward.
If the post is still intact and you are removing it because of a crack or something like that you would need to look inside the rail holes to make sure the rails have not been nailed together ... Some contractors nail the rails together or nail through the post into the rails. This keeps things more rigid but it really screws up things down the line when a repair needs to be done.
Given the case you have nails in your rails you will need a sawzall to cut the nails, separating the rails.
Given the rails are now separated or nail free.. I would pull the post out of the hole.
Check the new post for length... Making sure your depth of the hole is deep enough to sink the length of the post without having to cut the post.
The easiest way to go with this is check the length of the old post. But it if were broken off there is some difficulty there as I am sure some of it has rotted away.
In this case... Take a string line, from the post on the right of the repair to the post on the left of the repair... Pull the line tight... The string line would give you your assumed height of the new post. From that line measure down to the bottom of the hole. If your hole to line is 6 foot 4 inches and your new post is 7 foot long you have two options,... Either cut 8 inches off the new post or you dig down another 8 inches to make it work.
I would suggest to dig down the 8 inches. This would assure your post is more secure once your done tampering your post in.
Place the post in the hole... On somewhat of an angle, laying your post to one side, try to insert the rails back into the post on one side... left or right does not matter as long as you do one side at a time.
Pushing the post as far away as possible allowing the true length of distance between the new post and the last post next to it to increase in distance...
Now, I would be assuming you would have a new post in the hole with the rails inserted in their holes on the either side...
The tricky part is getting those long rails into the post without having to cut them back in any way.
If you push that post away from the line of the fence the distance will increase, allowing you to have the opportunity to insert the rails into the post holes ..
Having someone to work with you would help greatly. The extra person would help hold the rails in from the opposite side of the post so they do not fall back out while you are working on the second set of rails.
Pushing the post away, trying your best not to break off the ends of the old rails... Sometimes instead of pushing the post away you can simply lift the new post out of the ground... high up, about to the point where the bottom of the post is even with the ground level ... Doing this you would definitely need a helper. He/she would hold the rails placing the bottom rails in the bottom holes first.. and after you insert the rails in the bottom holes you would begin to lower the post into the hole. Once lowering you would see the top rails will get closer to the holes in the new post. You would then have your helper line the top rails in line with the top holes... you continue to lower the post into the ground until all four rails are inside.
I would surely hold onto the post so it does not fall over. If it fell to one side of the post hole it may be just enough room for the rail to come back out. 
Lining up one post with the other and keeping a straight line of fencing is done the same way with the string line. Attach a line from the side of the post 2 or 3 posts to the right of the new post to the side of the post 2 or 3 posts to the left of the new post... where ever that line falls near the new post is the for the most part, your fence line. You can use a level to level the post to the line, but Split rail posts are not the most uniform. They typically have milled edges that are not square all the way around... A good eye ball would be just fine.
Back fill about 8 inches of soil, tamp in solid ... Back fill in another 8 inches of soil and tamp again... until the post hole is filled with all the soil around that you removed when digging.
If you want you can drive a 20 penny galvanized nail through the new post into the two rails...
For the most part this repair is very easy but quite the challenge. There is nothing worse than being hit in the head by the top rail while trying to work on the bottom rail.
That is why having a helper is so much easier ... and safer too.
Best of luck to you ...
Greg's Fence, NJ~
How to Replace Rails in a Split Rail Fence
Source: https://www.doityourself.com/forum/fences-posts-railings-gates/466879-replacing-post-two-rail-fence.html