I've planned trips to WDW in 2014 and 2015 (and will be there again in a matter of weeks!) specifically to see the Nationals play the Braves at WWoS. I can't answer your question about whether you can walk around the complex if you don't have tickets (you do need to have a ticket to the game to get into Champion Stadium), but I do know that there are a variety of counter service concessions within the stadium and a table service grill.
There is bus transportation, but it's kind of a pain. The only way to get to WWoS on the Disney bus system is from All Star Sports, Pop Century, or Caribbean Beach. Unless things have changed since March 2015, there are no buses to WWoS from the TTC, Disney Springs, or any of the parks. I learned the hard way last year that the buses only run every 45-60 minutes, and wound up over an hour late to the game, at which point all of the starters and major league players had been pulled, so we paid $30 to see AAA and AA players. If you are planning on doing the bus, I would recommend picking it up at All Star Sports because that route goes straight to WWoS, whereas the other route picks up at Pop, goes to Caribbean Beach and each of its 5-6 internal stops, before going on to WWoS. For this year's trip, we're already planning on just taking Uber to and from the game.