Is there any indication from the movie, or other official material, on how long the Rebellion spent studying the stolen Death Star plans before they found out about the exhaust port vulnerability?
According to Wookieepedia, there is a range of less than 16 days in which it could have happened. I don't know that I 100% agree with their estimation, but here's how they break it down.
- According do the Galaxywide Newsnets published by West End Games, the Dissolution of the Imperial Senate took place on 0 BBY (35:3:5 GrS)
- According to Star Wars: The Essential Atlas, The Battle of Dathomir took place on 8 ABY (43:3:21)
They posit that that means that The Battle of Yavin took place somewhere between 0 BBY (35:3:5) and 0 BBY (35:3:21)
To me, this is narrow-sighted, assuming that the Battle of Dathomir must have taken place on almost the exact anniversary of the Battle of Yavin. It's been years since I read The Courtship of Princess Leia, but I can't find any reference to the battle being specifically on that anniversary.
If you follow their logic, that's 16 days from the Dissolution until the actual battle, but one must also shave off time for the events aboard the Death Star and travel time from there to Yavin. Unfortunately, we all know that travel times in Star Wars are all screwed up.
A full timeline of the plan retrieval on Wookieepedia (assembled from both movie G-canon and EU C-canon) confirms that the rebel command did not have access to the plans until Millennium Falcon delivered R2-D2 to Yavin 4.
How long did Jan Dodonna and company have to study the plans? That's hard to say.
According to The Movie Trilogy Sourcebook, the analysis took at least several hours, since it involved Dodonna preparing to go to sleep after not finding a solution; and getting to the idea after telling a child a story about killing a dragon via a chink in its armor. Realistically, it should be somewhere between 8 and 48 hours, since I doubt he would have gone to sleep without trying for at least 8 hours, and unlikely to have been able to function without sleep for 3 days straight.
Please note that we don't have a good official date estimate: the best Wookieepedia can do for the battle itself is "35:3:5 and 35:3:21" (the former is dissolution of Imperial Senate that we know preceded the battle as Vader informed Leia about it after capturing Tantive IV, and the latter is subtracting 8 years from a Dathomir battle whose date is known and happened at least 8 years after Yavin). This is explored in detail in @phantom's answer, but doesn't help much to narrow things down (since we can't assume Dodonna went without sleep for a full 16 days).