"My elder sister"
I think you mean お姉 and お兄, which are pronounced おねえ and おあに, respectively, and usually followed by honorific suffixes, like さん or ちゃん.
You would use the honorific form when addressing your older sibling (if you wanted to be respectful) or when talking about someone else's sibling.
If you are talking to someone outside your group (not part of your family or close friends), then you would typically use humble forms when talking about your own family. So you would drop honorifics and speak more plainly. This lowers yourself compared with the other person, so it is a more polite way of speaking. Honorifics for them, humble speech for you (and your group).
Keep in mind, if you have a close relationship with your older sibling, you might drop honorifics, because politeness also creates distance. And if you have a BAD relationship, you might drop honorifics to be intentionally insulting. Aren't politeness levels fun? :-)
Just one small correction - the reading for お兄 when used in the phrase お兄さん is おにいさん not おあにさん.
Good catch. I remembered the phonetic shift for あね but forgot it for あに. Doh!
おに is what you call your おにいさん when he's acting badly. :-)
(おに「鬼」is a type of ugly monster).