This is a very good point you have made. However, I have a few things Iwish to poit out.

1, Anthony, I believe, was getting more in thouch with his Italian side as an adult. he seemed to be finding himself, and I believe that he knew for quite some time that he wanted to be a singer. He very well could have taken lessons, though, some are simply born with a gift.

2, it was quite a while before he told his father about the opera. I would guess almost a year, and, if Anthony already knew how to sing, which is very likely, then it is very possible that he could have got a good part in Cavaleria Rusticana. Though I still wouldn't be very surprised if the Corleone Family help a bit. perhaps, Michale was nervous for his son and figured if he wasn't going to be a lawyer, then he would assure that his son would have a good career in singing.

Still, very little of this is explained, and can be considered a plot flaw. We can't be expected to be splitting hairs like this...