You need to put down some tack oil for the asphalt to stick to another surface or else it will come loose. Contact your street department and local asphalt company and get you a bucket. It is the same stuff used on road projects sprayed through a row of nozzles behind a tank truck called a distributor. It is also the stuff you don't want to drive through, but many do, cause it sticks to your paint when driving around paving machines at road construction projects.
Temperature is a concern. You usually don't want to pave streets under 50 degrees. Contact your local asphalt company and rent a concrete type compactor to compact your ramp from a rental company.
It will probably crack away at the feathered edge to 0". Best way would be to chop out a 2" or so deep section of the street so you always keep at least a couple inches in depth of new asphalt at the edge. Are you using hot mix or cold mix?