1st base will most likely be manned by Allen Craig this coming season. Adams' best chance to become the starter there will likely come in 2014 when Carlos Beltran is gone allowing Craig to shift to right.
Of course, injuries would change Adams' chances.
There's also a good chance that Oscar Taveras gets right field in 2014 squeezing out Adams.
I wouldn't be at all surprised to see Adams traded at some point to fill a need, most likely at short if Furcal is hurt or after 2013 if no other viable option for filling short surfaces.