Question would the feat Storm-Lashed ....

[AsheItachi]

... be able to let you fire arrows thru the spell fickle winds (http://paizo.com/pathfinderRPG/prd/ultimateMagic/spells/fickleWinds.html)

Storm Lashed Feat (http://www.d20pfsrd.com/feats/general-feats/storm-lashed)

[Red Fel]

No. Read the feat: Storm-Lashed lays out three specific things that it does. In rain, your visibility is reduced by 1/4, not 1/2, and you take only -2 to Perception. In high winds, you are treated as one category larger, and halve any Perception penalty. Gain +2 on all saving throws against electricity.
That's all the feat does. Now look at Fickle Winds - it functions as Wind Wall, but without impeding those within it. And Wind Wall explicitly causes arrows and bolts to miss, regardless of the size of the launcher.

So, no. Storm-Lashed does not let you fire through your enemy's Fickle Winds. Note, however, that if Fickle Winds is cast by an ally, it's a non-issue - the target(s) of the spell can fire freely, unimpeded.

[weckar]

For completeness: You're only one category larger when determining wind effects.