I recently went to Yokohama and Tokyo for a holiday. We visited this Bird cafe in Asakusa. It's called "Tori no Iru Cafe". It's not exactly a "cafe"; you can't eat/drink in there. There are two rooms, one with owls and the other larger room has loads of small breeds of parrots that fly around freely. We spent an hour in there. It was fun but quite noisy as you'd expect. haha
I didn't spend that much time with the owls. They were pretty...um...intimidating.
You had the option to wear a "rain coat" to prevent the birds from pooping all over you. I don't know why you wouldn't take it. You could also buy a capsule that's filled with bird food.