The film takes some liberties, but the core of the story is 100% accurate. The lawyer, James Donovan, did take the initial case of Rudolf Abel, and did negotiate the swap of Abel for U2 pilot Gary Powers and US graduate student Frederic Pryor. Donovan wrote a book in 1964 about the saga, titled Strangers on a Bridge; The Case of Colonel Abel and Francis Gary Powers.
The US graduate student that was swapped, Frederic Pryor, is still alive. He had a long career as an economics professor, mostly at Swarthmore College, and he still has some role at Swarthmore.
I thought the movie was excellent, and posted as much a day or two after it came out. It's a little slow moving at times, but it does a great job of capturing the paranoia of the era, and how much of a day-to-day battler there was between the US and Russia, as well as capturing the feelings of a handful of people, like the aforementioned, that were caught up in the Cold War.
There are a number of excellent films slated to be released in the next 2 months, but Bridge of Spies should be a contender for a number of awards.