The second sentence doesn't make sense. If you want to express a time frame you'd use the first.
He has been running his own business since graduation.
This means he currently runs his own business, and he started running his own business at or very near graduation and is still running it today. This is emphasizing him running his business during the entire time-frame from graduation to present.
He has been running his own business after graduation.
This doesn't make sense. Has been running... implies that this is an ongoing activity and to complete that sentence you'd need a continuous time-frame.
- He has been running his own business for ten years.
- He has been running his own business since he was a child.
The simple non-progressive moment in time "after graduation" is not a suitable tense. That would be like saying He has been running his own business in January. this is a verb tense confusion.