After listening to the luscious renditions of “I Could Have Danced All Night” and “On the Street Where You Live” in Arena Stage’s often tunefully adept if dramatically inert revival of “My Fair Lady,” you may never feel worthy of singing them in the shower again. Manna Nichols and Nicholas Rodriguez, the Eliza Doolittle and Freddie Eynsford-Hill of director Molly Smith’s new production, apply to their vocal performances a show-tune shimmer that does thorough justice to two of the musical’s beloved standards.