An old dog can learn at the same capability as any young dog just as long as it isn't having any serious health issues not allowing it to.
And yes you can condition your cat and dog to be okay with eachother, especially depending on your dogs response to the cat, the best way you can condition this is to have your dog and cat seperate, while slowly giving exposure to one another while making the cat feel safe, by this I mean begin with a few minutes of your dog in a crate and your cat having freedom do to do as it pleases, if the cat loses some of its insecurity this can be fixed quicker as compared to if the dog is the issue, if you worry your dog will get your cat it's best to reverse this, by putting the cat and a crate and letting the dog get some interaction with the cat, slowly though, because you need to not let the dog kick into a "Prey Drive" (i.e getting aggressive, barking, growling, lunging of any sort). at that point IMMEDIATELY remove the dog from the situation.. I would love to inform you more on this but I don't know many details of your exact situation.. Hope that helps, but the best thing you can do for any behavior issues such as that is contact a local certified trainer or behaviorist
<-- CPDT-KA by APDT (Association of Professional Dog Trainers)