Without emulating the feelings of an Emma towards her Henry, [Anne] would have attended on Louisa with a zeal above the common claims of regard, for [Captain Wentworth's] sake; and she hoped he would not long be so unjust as to suppose she would shrink unnecessarily from the office of a friend.

I read this as saying that although Anne would not go to the extremes of Emma in the poem, she would take more than usual care of Louisa specifically because Louisa was special to Captain Wentworth. The second half of the sentence does add that Anne hopes CW won't think that she would be reluctant to do her duty as Louisa's friend out of jealousy of the relationship between Louisa and CW.