So the phrase "longing on the good old days" has no meaning by itself. The sentence can be broken up into three parts, "she looked back," "with longing," and, "on the good old days." "she looked back," by itself can mean to look backwards (i.e. Behind you), but here, because it says, "on the good old days" it means she's "looking back" in time to the good old days, in other words, she's reminiscing about the past, recalling fond memories. You can also "look back" to a time of hardship, it basically means you're remembering and thinking about the past. "with longing" here means with a desire to return to the point in time she's reminiscing about.