Ashling would be close enough.
A-shling, to me, sounds like an American trying to pronounce it. And I've never heard and irish person say OSH-ling...except maybe an English speaker with notions from Dublin 4. The Dublin 4 (or D4 as we call it) accent is the butt of a lot of jokes here in Ireland - like is pronounced loike, dart becomes dort, and I guess Aisling could be pronounced Oshling.
But I digress...
Here's a link with correct pronounciation of aisling for the three main dialects. They're all much the same.