Which is the better temporary fix?
 

The shingles and the metal drip edge on my second story roof are flush resulting in about an 1/8 inch overhanging the gutter. So quite a bit of rainwater is flowing behind the gutters and falling down right next to the house, washing out a flower bed and making a mess. I had two people out to look at it: one suggested running a strip of starter shingles under the existing shingles which he said would give about 7/8" overhang, the other suggested running a strip of aluminum under the existing drip edge and bending it down toward the gutter. Both estimates were about $300, so which would be the better choice? Both men agreed that the existing roof is very poorly done, and one noticed some rot along the eve due to the flashing not completely covering the wood.

If the roof is truly that bad then the best "fix" is to replace the room before more issues come up and it ends up cosing more.
I highly doubt there suggesting adding a a better drip cap under the old one. Far more likly it's going to be over the old one and further up under the shingles and longer so the water runs into the gutters.
I've had to do this before for someone that just could not afford to do it right and it worked ok.

I've slipped metal under the starter course to extend the overhang. As joecaption said, not the preferred method but it worked.