I dont think he was in so much awe as he was just around Keith all the time. Ringo and Keith hung out so naturaly Zakk was around too. Zakk said that Ringo would never give him any lessons and then one day Moonie gave him a set of drums and might have shown him a few things, but that was it. I think Zakk learned mostly by listening to records.
I believe this is so, but i'll have to go and find the story now to back it up.