You're thinking of a foal not a pony. A pony is a full grown horse, only smaller.
You would start training a warhorse almost from birth. Leading it, handling it, teaching it not to be afraid. That would be the biggest problem training a warhorse. Most horses are very nervous and it would take a lot of work to keep your horse from freaking out when a dozen screaming crazed goblins charged you waving axes around.
You could probably start riding the horse around 3 years old, but it would be best to wait until 4 years before you load him up with heavy armor.
If I was your player I would buy a 3-4 year old horse. It would grow up faster and you could still train it to do whatever you want. The price of an untrained horse should be much lower than a warhorse. I'm not sure how long it would take to turn an untrained horse into a warhorse, but in game terms several months should work. It should be ok to ride after about 30 days of training, but I wouldn't take it into combat.
You could role-play out one of the horse training scenes. I did something like that last year and my players loved it. They were trying to ride a dire tiger though.

We had a little rodeo that lasted most of a session.