need advise on the easiest and least expensive way on putting a roof on a free standing 12x12 deck/patio.
I think the asphalt shingles would be the way to go. For such a small area the cost would be minimal.
Shingles are not a roof. They cover a roof.
Easiest way will depend on your Carpenter skill level. A slanted roof would be less difficult than pitched. 4 x 4 corner posts will be needed and then they will need to be braced together in the shape of a frame. Then you will need the roof framed with support beams. A lot of folks seem to like that metal tin for a barn look. But wooden shingles would look really neat overlapped on top of plywood.