How to start reverse engineering a circuit?
The P8255A has no internal program storage- any "programming" is done externally, so you don't need to worry about that- the program in the PC is where the configuration is done. 

As far as reverse engineering an old card like that, it should be possible. If it's an edge-connector ISA card, it won't be particularly cheap to have a new card made. You might be breaking some copyright rules by using the software with your own card, but I'll let you investigate that (non-technical) issue. There used to be perf board available with an ISA edge connector pattern on it. With only 8 ICs it could easily be wired up with magnet wire on a perf board in an hour or three, but it would be easy to miss something and have a heck of a time troubleshooting it. 

Edit: 

On your board they didn't bother getting either solder mask or gold plating on the edge connector contacts. Pretty infra-dig but I guess it worked for a while. Even gold flash would be better. 

Here's the Vector prototyping board #4613-1, Digikey still has 15 in stock (but they'll not be re-ordering).