One of the most common and inexpensive means of finishing a flat roof rolled asphalt installs over a drip edge flashing that prevents water from getting under the rolled asphalt material around.
Flat roof with sheet flow off edge.
Then reshingle sliding metal step flashing behind the fascia board the trim behind the gutter.
Rubber roof repair kit available on amamzon.
Install roof drip edge along the eaves add underlayment then place drip edge along the gable end.
Slip the wide flat flange of the material all the way under the first course of roof shingles as illustrated.
If it doesn t pull the drip edge out slightly from under the shingles.
The best way is to install the roof drip edge only along the eaves first then place ice and water barrier in the snowbelt or felt paper underlayment over the drip edge.
It bends around the edge of the roof channeling any moisture that penetrates the shingles away from the roof decking.
Secure the drip edge with roofing cement.
Make sure the outer edge extends over the gutter.
Depending on how the roofs join you may have to cut a slot to work it in far enough.
This should cover the most leak prone areas.