I am looking for helpful advice about which feat is best to select at 1st level for Ysoki Outlaw Mechanic build
Exocortex or drone?
Exocortex or drone?
Drone
Longarm proficiency is good; small arms are terrible enough and grenades expensive enough that you don't want to use either primarily. Sniper weapon proficiency does even more damage, but there's a smaller selection and while unwieldy won't normally be a problem for you it will come up occasionally. If you plan to use your drone as your main attack form rather than get a standard action attack from both you and your drone, neither of those is useful of course.
Improved initiative, or iron will to shore up your weak save are both worthwhile.
I'm assuming that melee is a bad place for you. Mobility makes it easier to escape melee and could lead to shot on the run. Sidestep is another way to escape melee, skill synergy (acrobatics and perhaps stealth) helps with a third way.
I'd probably get longarm proficiency; YMMV.
If you picked up slight of hand from outlaw, might as well take skill versatility to add Stealth and Acrobatics as class skills.
Although Longarms basically doubles your damage.
I think Longarm is probably your best bet, but you may not be able to take advantage of it when you start play. Buying even a half decent longarm for both yourself and your drone is ~half your starting money. You can do it, but it'll be tight and you'll be buying cheap longarms that you'll replace as soon as you can.
What I'd do is take Longarm and buy the best longarm or advanced two hander for my combat drone I could afford and only buy me a cheap melee weapon. Then I'd loot the first longarm I came across and use that as my personal weapon.