It is possible to run some of the Big Fish games but it must be done via WINE(can be found in the Ubuntu Software Center) -WINE-64bit or Codeweavers Crossover Linux
The particular game that you are talking about would need to be checked with the application database on either site.
WINE is free but is mostly unsupported; only blog like answers by users and the WINE team(the last is limited as they are constantly working on WINE itself and not available to give personal support). To run most windows programs you will need to install other supporting programs via winetricks(will be installed with WINE if installed via the Software Center) programs like Play-on-Linux are supposed to make it easier to install the needed programs.
Quite frankly I gave up trying to get WINE to work, the instructions needed to get them running were very hard for me to follow as the answers were in the middle of blog like posts that are a back and forth discussion between all the participants, I lost the information I needed in all the back and forth. This was just me, many people use WINE successfully so they must be able to follow it.
I found success with Crossover. It is not a free program but includes email support and will install alot of the needed extra programs automatically, so you are getting more than the free version of WINE. It is a annual subscription, $60 for the initial year and then it can be renewed for $30 a year. You can try is for 2 weeks for free to see if you like it, but it will sotp working after the trial if you don't but it. They often run specials so you can sometimes get it for cheaper than the $60 price but you would kind of have to get lucky, they don't run specials on any schedule that I know of. Note you can get crossover via the Ubuntu Software CEnter but I would recommend that you download the .deb file directly form the Codeweavers' website, it is usually more up to date and updating is easier
The short answer here is that there is no easy & free way to play Windows game or run Windows programs on Ubuntu but it can be done easy(relatively) or free. Even with WINE's free version or Codeweavers' value added version, not all programs or games will run on Linux and those that do may not run as well, though some run better.
As I said above you would need to check the application database to see if and how well you particular game will run at all and if so how well. Both databases use a medal system to rate the programs. WINE rates their programs Platunium-Garbage. Codeweavers uses Gold-Known not to work. I will refer you to the respective website for the exact rating system as this answer has almost become a novel and it is much easier for you to reading it yourself then trying to understand my paraphrasing.
I hope this long and rambling answer was helpful. Sorry it was so long.