Who is that song written about?

According to this interview with John Oates, it's about Frankie Valli (sort of):

http://beyondrace.com/columns/catching-up/891-catching-up-with-john-oates

I loved “Back Together Again.” Was that about Frankie Valli?

Yes, about Frankie Valli. It wasn't literally [about him]. We were coming back from somewhere, first class, and realized Frankie Valli was right next to me. We talked. I was in my prime and he wasn't. To hear his enthusiasm and still having passion to sing and perform was great. And I kinda took the feeling from that conversation and put it into that song. It's not like about getting back together [romantically]. More just about keeping it together personally.