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If that is the correct way to say it, then I like it very much. More than the Anya spelling.
I love the earthy, warm feel (and the Russian-ness) of Anya, esp as a pet name for an Anna. Aine has a lovely line to it, which seems more suiting if used as a given name. I'm not certain that made sense. It's incredible how a small change in spelling call alter the feel of a name.
Awn-ya or On-ya is the correct pronounciation.
On-ya and Anya sound about the same in my accent. Awn-ya isn't much off in my accent, either- just enlongating the first syllable.