Can I use Rigid Foam Insulation instead of fiber insulation?
Yes. There are several types of rigid foam board. Polyisocyanurate (polyiso) has the best R-value at 6.5 (after aging). A 2 inch thick board will be at least R-12.9.  

There are two main methods you can use from inside the room. With the first method, sometimes called "Cut-and-Cobble", you cut the panels to fit inside the joists and then use spray foam (e.g. Great Stuff) around each panel to create a tight air seal. If you have a 2x4 exterior wall, you could even get 3 inches of foam in there if you can find 3" panels or use a 2" and 1" panel together. I recently did exactly this myself in a bedroom that I gutted down to the studs.

Another method is to add the foam panels to the inside of the wall on top of the studs, tape the panels together with foil tape, and then put the drywall over the panels. This second method is preferred since it reduces thermal bridging from the cold studs into the wall and also provides a continuous air seal across the entire wall. If you do put the panels over the studs, you must fasten them to the studs and then use longer drywall fasteners to connect to the studs since the foam is not considered structural. The downsides to this methods are that all of your trim and receptacles must account for the additional wall thickness and you lose a bit of space in the room, depending on foam thickness.