What He's Done to Get Picked up in Fantasy:
Pitch very strongly in three consecutive starts (after struggling in his debut). Currently, the owner of a shiny 2.38 ERA & 1.06 WHIP.
What to Expect Going Forward:
One area that needs to be taken into account is that his last three starts have been made in notoriously pitching friendly parks (San Diego, San Francisco & New York). Of course this doesn't mean that advantage is going away soon as Hahn is going to have a head start on ERA versus similar pitchers as he gets to pitch half of his games at Petco Park (very awesome stadium by the way).
If I would have looked at his track record upon his call up, I would have been a bit skeptical as he got called up straight from AA (similar to my skepticism regarding Marco Gonzales).
In this case, he does have the Petco advantage and while he has gotten lucky on the BABIP side of things (.255) it's not an uber-extreme advantage. Assuming a BABIP of .300 the rest of the way (1.17 higher than .255), this factor multiplied by his current ERA would place him at 2.8 which is between his FIP and xFIP totals.
One factor that could favor him keeping his ERA very low for the rest of the year as he has been a bit unlucky in giving up homers (15.4% HR/FB ratio). If this total comes down this could counteract any side effects from BABIP related regression (though I find it a bit ambitious for the 10.72 K/9 to hold up for the rest of the season).
The Padres have definitely struggled (especially on the offense side) and are currently 34-44 and playing for the future. It has been so bad that it has already cost their former General Manager Josh Byrnes his job and who knows whether this season will cost Bud Black his job too. It is clear that the Padres have hit the throw stuff at the wall and see what sticks stage of the season and based on the early returns it appears they found a winner in Jesse Hahn. The Padres may be beyond hope of getting themselves into contention but Hahn appears to have the stuff to help fantasy owners keep their teams in contention all summer long.
All numbers, statistics, etc. found on either fangraphs.com or baseball-reference.com.