How to access music when on airplane mode?

shouldn't be a problem. Airplane mode only keeps you cellular data from connecting. You should still be able to connect to wi-fi as well.

Is this music that you have store locally on your iPhone? Or is it music you're streaming?

Same issue with me on my new iPhone 5s. With airplane mode off, all my music is available. With airplane mode switched on, I see only about 8-10 recently purchased songs. I did not have this issue with my previous iPhone 5. ***?