Will an old CPU work well on a new computer?

Yes the Intel i5-2400 will work fine with your new computer, don't buy new one unless you need new one.

Definitely use your old cpu. It is a much better and faster cpu then the fx 4300. You will be wasting money buying that fx 4300. You may want to pick up a better motherboard for you old cpu since I'm guessing you bought your last pc prebuilt. You can use the money you save for a better graphics card which will give you a good performance boost over the built you already have listed. I don't know how much knowledge you have of pc part, I just want to point out a mistake people make when they look at cpu's specs. People think the higher the clock speed of cpu the faster it is, which is true and untrue. It's only true when you comparing cpu's of the same architecture. For example, fx-6300 3.5ghz vs fx-6350 3.9ghz both are the same cpu architecture so the higher clock speed one will be faster. When you compare AMD to Intel it's a whole different story, Intel is more efficient CPU's so a lower clock Intel are faster then the less efficient AMD cpu's. AMD usually use higher clock speed then Intel to make up for what they lack in efficiency but still the i5 is a better gaming cpu even compare to the higher end AMD cpu's.

So that may be more info then what you needed but just wanted to back up what I said.

So use that money towards the GPU. Check to see if the pc the i5 is in has pci-e x16 slot in it. If you have problems finding that comment the model of the pc and I will check it for you. If it does not you can pick up a motherboard for about $60.

Here a link to backup what I said. http://www.tomshardware.com/reviews/gaming-cpu-review-overclock,3106-5.html

Realize that those two CPUs need different sockets, which means different motherboards.
At the following site, the two CPUs (and a bunch of others) are compared using the futuremark benchmarks. The FX ranks 46, the i5 69.

http://www.futuremark.com/hardware/cpu

Get the Game Recommended (or Optimal) specs for the games you want to play. Then compare what the game(s) want to what hardware you have.
Example:
Skyrim wants…
-Windows 7/Vista/XP PC (32 or 64 bit)
-Quad-core Intel or AMD CPU processor
-4GB System RAM
-6GB free HDD (Hard disk drive) space
-DirectX 9.0c compatible NVIDIA or AMD ATI video card with 1GB of RAM: Nvidia GeForce GTX 260 or higher; ATI Radeon HD 4890 or higher
-DirectX compatible sound card
-Internet access for Steam activation

for future games you will want to get better hardware.

If you already have the i5 the FX 4300 would not be an improvement.