Here are the several ways available for cooling your house.
1. Extending the duct work
Connecting the existing duct work to the additional room might be a smart choice. But you must ensure that your AC unit will be able to support the additional square foot area.
If you extend the ducts but your AC unit is too small it will affect the comfort of whole house.
2. Installing a window air conditioner
Instead of extending the duct work there is an easier choice for you, i.e. to install a window air conditioner. Window air conditioners are most common and are highly useful to cool one room.
Installing a window air conditioner is simple and also low cost. However this air conditioner does not look good as they are big in size and make loud noise. For installing this unit you will require an exterior wall.
This is the best option for you if your additional room will not be in much of the use as it is less expensive and easiest to install.